The Askey-scheme of hypergeometric orthogonal polynomials
and its q -analogue
Roelof Koekoek
Rene F. Swarttouw
February 20, 1996
Abstract
We list the so-called Askey-scheme of hypergeometric orthogonal polynomials. In chapter 1 we
give the denition, the orthogonality relation, the three term recurrence relation and generating
functions of all classes of orthogonal polynomials in this scheme. In chapter 2 we give all limit
relations between dierent classes of orthogonal polynomials listed in the Askey-scheme.
In chapter 3 we list the q-analogues of the polynomials in the Askey-scheme. We give their
denition, orthogonality relation, three term recurrence relation and generating functions. In
chapter 4 we give the limit relations between those basic hypergeometric orthogonal polynomials.
Finally, in chapter 5 we point out how the `classical' hypergeometric orthogonal polynomials of
the Askey-scheme can be obtained from their q-analogues.
Acknowledgement
We would like to thank Professor Tom H. Koornwinder who suggested us to write a report like
this. He also helped us solving many problems we encountered during the research and provided
us with several references.
Contents
Preface
Denitions and miscellaneous formulas
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Introduction : : : : : : : : : : : : : : : : : : : : : : : : :
The q-shifted factorials : : : : : : : : : : : : : : : : : : :
The q-gamma function and the q-binomial coecient : :
Hypergeometric and basic hypergeometric functions : :
The q-binomial theorem and other summation formulas
Transformation formulas : : : : : : : : : : : : : : : : : :
Some special functions and their q-analogues : : : : : :
The q-derivative and the q-integral : : : : : : : : : : : :
ASKEY-SCHEME
1 Hypergeometric orthogonal polynomials
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2 Limit relations between hypergeometric orthogonal polynomials
2.1 Wilson ! Continuous dual Hahn : : : : : : : : : : : : : : : : : : : : : :
2.2 Wilson ! Continuous Hahn : : : : : : : : : : : : : : : : : : : : : : : : :
2.3 Wilson ! Jacobi : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
2.4 Racah ! Hahn : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
2.5 Racah ! Dual Hahn : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
2.6 Continuous dual Hahn ! Meixner-Pollaczek : : : : : : : : : : : : : : : :
2.7 Continuous Hahn ! Meixner-Pollaczek : : : : : : : : : : : : : : : : : :
2.8 Continuous Hahn ! Jacobi : : : : : : : : : : : : : : : : : : : : : : : : :
2.9 Hahn ! Jacobi : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
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1
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1.9
1.10
1.11
1.12
1.13
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1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
Wilson : : : : : : : : : : : : : : : : :
Racah : : : : : : : : : : : : : : : : :
Continuous dual Hahn : : : : : : : :
Continuous Hahn : : : : : : : : : : :
Hahn : : : : : : : : : : : : : : : : : :
Dual Hahn : : : : : : : : : : : : : :
Meixner-Pollaczek : : : : : : : : : :
Jacobi : : : : : : : : : : : : : : : : :
1.8.1 Gegenbauer / Ultraspherical
1.8.2 Chebyshev : : : : : : : : : :
1.8.3 Legendre / Spherical : : : : :
Meixner : : : : : : : : : : : : : : : :
Krawtchouk : : : : : : : : : : : : : :
Laguerre : : : : : : : : : : : : : : : :
Charlier : : : : : : : : : : : : : : : :
Hermite : : : : : : : : : : : : : : : :
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5
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2.10
2.11
2.12
2.13
2.14
2.15
2.16
2.17
2.18
2.19
2.20
2.21
2.22
2.23
Hahn ! Meixner : : : : : : : :
Hahn ! Krawtchouk : : : : : :
Dual Hahn ! Meixner : : : : :
Dual Hahn ! Krawtchouk : :
Meixner-Pollaczek ! Laguerre
Meixner-Pollaczek ! Hermite :
Jacobi ! Laguerre : : : : : : :
Jacobi ! Hermite : : : : : : :
Meixner ! Laguerre : : : : : :
Meixner ! Charlier : : : : : :
Krawtchouk ! Charlier : : : :
Krawtchouk ! Hermite : : : :
Laguerre ! Hermite : : : : : :
Charlier ! Hermite : : : : : :
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q-SCHEME
3 Basic hypergeometric orthogonal polynomials
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
3.20
3.21
3.22
3.23
3.24
3.25
3.26
3.27
3.28
3.29
Askey-Wilson : : : : : : : : : : : : : : : : : :
q-Racah : : : : : : : : : : : : : : : : : : : : :
Continuous dual q-Hahn : : : : : : : : : : : :
Continuous q-Hahn : : : : : : : : : : : : : : :
Big q-Jacobi : : : : : : : : : : : : : : : : : : :
3.5.1 Big q-Legendre : : : : : : : : : : : : :
q-Hahn : : : : : : : : : : : : : : : : : : : : : :
Dual q-Hahn : : : : : : : : : : : : : : : : : :
Al-Salam-Chihara : : : : : : : : : : : : : : : :
q-Meixner-Pollaczek : : : : : : : : : : : : : :
Continuous q-Jacobi : : : : : : : : : : : : : :
3.10.1 Continuous q-ultraspherical / Rogers :
3.10.2 Continuous q-Legendre : : : : : : : : :
Big q-Laguerre : : : : : : : : : : : : : : : : :
Little q-Jacobi : : : : : : : : : : : : : : : : :
3.12.1 Little q-Legendre : : : : : : : : : : : :
q-Meixner : : : : : : : : : : : : : : : : : : : :
Quantum q-Krawtchouk : : : : : : : : : : : :
q-Krawtchouk : : : : : : : : : : : : : : : : : :
Ane q-Krawtchouk : : : : : : : : : : : : : :
Dual q-Krawtchouk : : : : : : : : : : : : : : :
Continuous big q-Hermite : : : : : : : : : : :
Continuous q-Laguerre : : : : : : : : : : : : :
Little q-Laguerre / Wall : : : : : : : : : : : :
q-Laguerre : : : : : : : : : : : : : : : : : : : :
Alternative q-Charlier : : : : : : : : : : : : :
q-Charlier : : : : : : : : : : : : : : : : : : : :
Al-Salam-Carlitz I : : : : : : : : : : : : : : :
Al-Salam-Carlitz II : : : : : : : : : : : : : : :
Continuous q-Hermite : : : : : : : : : : : : :
Stieltjes-Wigert : : : : : : : : : : : : : : : : :
Discrete q-Hermite I : : : : : : : : : : : : : :
Discrete q-Hermite II : : : : : : : : : : : : : :
2
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45
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4 Limit relations between basic hypergeometric orthogonal polynomials
4.1 Askey-Wilson ! Continuous dual q-Hahn : : : : : : : : : : : : : : : : : : : : :
4.2 Askey-Wilson ! Continuous q-Hahn : : : : : : : : : : : : : : : : : : : : : : : :
4.3 Askey-Wilson ! Big q-Jacobi : : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.4 Askey-Wilson ! Continuous q-Jacobi : : : : : : : : : : : : : : : : : : : : : : :
4.5 Askey-Wilson ! Continuous q-ultraspherical / Rogers : : : : : : : : : : : : : :
4.6 q-Racah ! Big q-Jacobi : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.7 q-Racah ! q-Hahn : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.8 q-Racah ! Dual q-Hahn : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.9 q-Racah ! q-Krawtchouk : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.10 q-Racah ! Dual q-Krawtchouk : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.11 Continuous dual q-Hahn ! Al-Salam-Chihara : : : : : : : : : : : : : : : : : : :
4.12 Continuous q-Hahn ! q-Meixner-Pollaczek : : : : : : : : : : : : : : : : : : : :
4.13 Big q-Jacobi ! Big q-Laguerre : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.14 Big q-Jacobi ! Little q-Jacobi : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.15 Big q-Jacobi ! q-Meixner : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.16 q-Hahn ! Little q-Jacobi : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.17 q-Hahn ! q-Meixner : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.18 q-Hahn ! Quantum q-Krawtchouk : : : : : : : : : : : : : : : : : : : : : : : : :
4.19 q-Hahn ! q-Krawtchouk : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.20 q-Hahn ! Ane q-Krawtchouk : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.21 Dual q-Hahn ! Ane q-Krawtchouk : : : : : : : : : : : : : : : : : : : : : : : :
4.22 Dual q-Hahn ! Dual q-Krawtchouk : : : : : : : : : : : : : : : : : : : : : : : :
4.23 Al-Salam-Chihara ! Continuous big q-Hermite : : : : : : : : : : : : : : : : : :
4.24 Al-Salam-Chihara ! Continuous q-Laguerre : : : : : : : : : : : : : : : : : : : :
4.25 q-Meixner-Pollaczek ! Continuous q-ultraspherical / Rogers : : : : : : : : : :
4.26 Continuous q-Jacobi ! Continuous q-Laguerre : : : : : : : : : : : : : : : : : :
4.27 Continuous q-ultraspherical / Rogers ! Continuous q-Hermite : : : : : : : : :
4.28 Big q-Laguerre ! Little q-Laguerre / Wall : : : : : : : : : : : : : : : : : : : : :
4.29 Big q-Laguerre ! Al-Salam-Carlitz I : : : : : : : : : : : : : : : : : : : : : : : :
4.30 Little q-Jacobi ! Little q-Laguerre / Wall : : : : : : : : : : : : : : : : : : : : :
4.31 Little q-Jacobi ! q-Laguerre : : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.32 Little q-Jacobi ! Alternative q-Charlier : : : : : : : : : : : : : : : : : : : : : :
4.33 q-Meixner ! q-Laguerre : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.34 q-Meixner ! q-Charlier : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.35 q-Meixner ! Al-Salam-Carlitz II : : : : : : : : : : : : : : : : : : : : : : : : : :
4.36 Quantum q-Krawtchouk ! Al-Salam-Carlitz II : : : : : : : : : : : : : : : : : :
4.37 q-Krawtchouk ! Alternative q-Charlier : : : : : : : : : : : : : : : : : : : : : :
4.38 q-Krawtchouk ! q-Charlier : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.39 Ane q-Krawtchouk ! Little q-Laguerre / Wall : : : : : : : : : : : : : : : : :
4.40 Dual q-Krawtchouk ! Al-Salam-Carlitz I : : : : : : : : : : : : : : : : : : : : :
4.41 Continuous big q-Hermite ! Continuous q-Hermite : : : : : : : : : : : : : : : :
4.42 Continuous q-Laguerre ! Continuous q-Hermite : : : : : : : : : : : : : : : : :
4.43 q-Laguerre ! Stieltjes-Wigert : : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.44 Alternative q-Charlier ! Stieltjes-Wigert : : : : : : : : : : : : : : : : : : : : :
4.45 q-Charlier ! Stieltjes-Wigert : : : : : : : : : : : : : : : : : : : : : : : : : : : :
4.46 Al-Salam-Carlitz I ! Discrete q-Hermite I : : : : : : : : : : : : : : : : : : : : :
4.47 Al-Salam-Carlitz II ! Discrete q-Hermite II : : : : : : : : : : : : : : : : : : : :
3
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5 From basic to classical hypergeometric orthogonal polynomials
5.1 Askey-Wilson ! Wilson : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
5.2 q-Racah ! Racah : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
5.3 Continuous dual q-Hahn ! Continuous dual Hahn : : : : : : : : : : : : : : : :
5.4 Continuous q-Hahn ! Continuous Hahn : : : : : : : : : : : : : : : : : : : : : :
5.5 Big q-Jacobi ! Jacobi : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
5.5.1 Big q-Legendre ! Legendre / Spherical : : : : : : : : : : : : : : : : : :
5.6 q-Hahn ! Hahn : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
5.7 Dual q-Hahn ! Dual Hahn : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
5.8 Al-Salam-Chihara ! Meixner-Pollaczek : : : : : : : : : : : : : : : : : : : : : :
5.9 q-Meixner-Pollaczek ! Meixner-Pollaczek : : : : : : : : : : : : : : : : : : : : :
5.10 Continuous q-Jacobi ! Jacobi : : : : : : : : : : : : : : : : : : : : : : : : : : :
5.10.1 Continuous q-ultraspherical / Rogers ! Gegenbauer / Ultra-spherical :
5.10.2 Continuous q-Legendre ! Legendre / Spherical : : : : : : : : : : : : : :
5.11 Big q-Laguerre ! Laguerre : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
5.12 Little q-Jacobi ! Jacobi : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
5.12.1 Little q-Legendre ! Legendre / Spherical : : : : : : : : : : : : : : : : :
5.12.2 Little q-Jacobi ! Laguerre : : : : : : : : : : : : : : : : : : : : : : : : :
5.13 q-Meixner ! Meixner : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
5.14 Quantum q-Krawtchouk ! Krawtchouk : : : : : : : : : : : : : : : : : : : : : :
5.15 q-Krawtchouk ! Krawtchouk : : : : : : : : : : : : : : : : : : : : : : : : : : : :
5.16 Ane q-Krawtchouk ! Krawtchouk : : : : : : : : : : : : : : : : : : : : : : : :
5.17 Dual q-Krawtchouk ! Krawtchouk : : : : : : : : : : : : : : : : : : : : : : : : :
5.18 Continuous big q-Hermite ! Hermite : : : : : : : : : : : : : : : : : : : : : : :
5.19 Continuous q-Laguerre ! Laguerre : : : : : : : : : : : : : : : : : : : : : : : : :
5.20 Little q-Laguerre / Wall ! Laguerre : : : : : : : : : : : : : : : : : : : : : : : :
5.20.1 Little q-Laguerre / Wall ! Charlier : : : : : : : : : : : : : : : : : : : :
5.21 q-Laguerre ! Laguerre : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
5.21.1 q-Laguerre ! Charlier : : : : : : : : : : : : : : : : : : : : : : : : : : : :
5.22 Alternative q-Charlier ! Charlier : : : : : : : : : : : : : : : : : : : : : : : : : :
5.23 q-Charlier ! Charlier : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
5.24 Al-Salam-Carlitz I ! Charlier : : : : : : : : : : : : : : : : : : : : : : : : : : : :
5.24.1 Al-Salam-Carlitz I ! Hermite : : : : : : : : : : : : : : : : : : : : : : :
5.25 Al-Salam-Carlitz II ! Charlier : : : : : : : : : : : : : : : : : : : : : : : : : : :
5.25.1 Al-Salam-Carlitz II ! Hermite : : : : : : : : : : : : : : : : : : : : : : :
5.26 Continuous q-Hermite ! Hermite : : : : : : : : : : : : : : : : : : : : : : : : : :
5.27 Stieltjes-Wigert ! Hermite : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
5.28 Discrete q-Hermite I ! Hermite : : : : : : : : : : : : : : : : : : : : : : : : : :
5.29 Discrete q-Hermite II ! Hermite : : : : : : : : : : : : : : : : : : : : : : : : : :
Bibliography
Index
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121
Preface
This report deals with orthogonal polynomials appearing in the so-called Askey-scheme of hypergeometric orthogonal polynomials and their q-analogues. Most formulas listed in this report can
be found somewhere in the literature, but a handbook containing all these formulas did not exist.
We collected known formulas for these hypergeometric orthogonal polynomials and we arranged
them into the Askey-scheme and into a q-analogue of this scheme which we called the q-scheme.
This q-scheme was not completely documented in the literature. So we lled in some gaps in order
to get some sort of `complete' scheme of q-hypergeometric orthogonal polynomials.
In chapter 0 we give some general denitions and formulas which can be used to transform
several formulas into dierent forms of the same formula. In the other chapters we used the most
common notations, but sometimes we had to change some notations in order to be consistent.
For each family of orthogonal polynomials listed in this report we give the conditions on
the parameters for which the corresponding weight function is positive. These conditions are
mentioned in the orthogonality relations. We remark that many of these orthogonal polynomials
are still polynomials for other values of the parameters and that they can be dened for other
values as well. That is why we gave no restrictions in the denitions. As pointed out in chapter 0
some denitions can be transformed into dierent forms so that they are valid for some values of
the parameters for which the given form has no meaning. Other formulas, such as the generating
functions, are only valid for some special values of parameters and arguments. These conditions
are left out in this report.
We are aware of the fact that this report is by no means a full description of all that is known
about (basic) hypergeometric orthogonal polynomials. More on each listed family of orthogonal
polynomials can be found in the articles and books to which we refer.
In later versions of this report we want to add recurrence relations for the monic polynomials
in each case, id est for the polynomials with leading coecient equal to 1. We also hope to include
more formulas containing quadratic transformations and we want to pay more attention to the
transformation q $ q 1.
Comments on this version of the report and suggestions for improvement are most welcome.
If you nd errors or gaps or if you have suggestions for inclusion of more formulas on (basic)
hypergeometric orthogonal polynomials, please contact us and let us know.
Roelof Koekoek and Rene F. Swarttouw.
Roelof Koekoek
Delft University of Technology
Faculty of Technical Mathematics and Informatics
Mekelweg 4
2628 CD Delft
The Netherlands
[email protected]
5
Rene F. Swarttouw
Free University of Amsterdam
Faculty of Mathematics and Informatics
De Boelelaan 1081
1081 HV Amsterdam
The Netherlands
[email protected]
6
Denitions and miscellaneous
formulas
0.1 Introduction
In this report we will list all known sets of orthogonal polynomials which can be dened in
terms of a hypergeometric function or a basic hypergeometric function.
In the rst part of the report we give a description of all classical hypergeometric orthogonal
polynomials which appear in the so-called Askey-scheme. We give denitions, orthogonality relations, three term recurrence relations, dierential or dierence equations and generating functions
for all families of orthogonal polynomials listed in this Askey-scheme of hypergeometric orthogonal
polynomials.
In the second part we obtain a q-analogue of this scheme. We give denitions, orthogonality
relations, three term recurrence relations, dierence equations and generating functions for all
known q-analogues of the hypergeometric orthogonal polynomials listed in the Askey-scheme.
Further we give limit relations between dierent families of orthogonal polynomials in both
schemes and we point out how to obtain the classical hypergeometric orthogonal polynomials from
their q-analogues.
The theory of q-analogues or q-extensions of classical formulas and functions is based on the
observation that
1 q = :
lim
q!1 1 q
Therefore the number (1 q )=(1 q) is sometimes called the basic number [].
Now we can give a q-analogue of the Pochhammer-symbol (a)k which is dened by
(a)0 := 1 and (a)k := a(a + 1)(a + 2) (a + k 1); k = 1; 2; 3; : : ::
This q-extension is given by
(a; q)0 := 1 and (a; q)k := (1 a)(1 aq)(1 aq2) (1 aqk 1); k = 1; 2; 3; : :::
It is clear that
(q ; q)k = () :
lim
k
q!1 (1 q)k
In this report we will always assume that 0 < q < 1.
For more details concerning the q-theory the reader is referred to the book [114] by G. Gasper
and M. Rahman.
Since many formulas given in this report can be reformulated in many dierent ways we will
give a selection of formulas , which can be used to obtain other forms of denitions, orthogonality
relations and generating functions.
Most of these formulas given in this chapter can be found in [114].
We remark that in orthogonality relations we often have to add some condition(s) on the
parameters of the orthogonal polynomials involved in order to have positive weight functions. By
7
using the famous theorem of Favard these conditions can also be obtained from the three term
recurrence relation.
In some cases, however, some conditions on the parameters may be needed in other formulas
too. For instance, the denition (1.11.1) of the Laguerre polynomials has no meaning for negative
integer values of the parameter . But in fact the Laguerre polynomials are also polynomials in
the parameter . This can be seen by writing
n
X
L(n) (x) = n1! ( kn!)k ( + k + 1)n k xk :
k=0
In this way the Laguerre polynomials are dened for all values of the parameter .
A similar remark holds for the Jacobi polynomials given by (1.8.1). We may also write (see
section 0.4 for the denition of the hypergeometric function 2 F1)
Pn(;)(x) = ( 1)n ( +n!1)n 2 F1 n; n + + +1 + 1 1 +2 x ;
which implies the well-known symmetry relation
Pn(;)(x) = ( 1)n Pn(;)( x):
Even more general we have
Pn;)(x) =
(
n ( n)
1 x k:
1X
k (n + + + 1) ( + k + 1)
k
n k
n! k=0 k!
2
From this form it is clear that the Jacobi polynomials can be dened for all values of the parameters
and although the denition (1.8.1) is not valid for negative integer values of the parameter .
We will not indicate these diculties in each formula.
Finally, we remark that in each recurrence relation listed in this report, except for (1.8.24) for
the Chebyshev polynomials of the rst kind, we may use P 1(x) = 0 and P0(x) = 1 as initial
conditions.
0.2 The q-shifted factorials
The symbols (a; q)k dened in the preceding section are called q-shifted factorials. They can
also be dened for negative values of k as
(a; q)k := (1 aq 1)(1 aq1 2 ) (1 aqk ) ; a 6= q; q2; q3 ; : : :; q k ; k = 1; 2; 3; : : :: (0.2.1)
Now we have
1 n
n
(a; q) n = (aq 1n ; q) = ((qaqa1 ; q)) q( 2 ) ; n = 0; 1; 2; : : :;
(0.2.2)
where
We can also dene
This implies that
n
n
n = 1 n(n 1):
2
2
(a; q)1 =
1
Y
(1 aqk ):
k=0
(a; q)1 ;
(a; q)n = (aq
n; q)
1
8
(0.2.3)
and, for any complex number ,
(a; q)1 ;
(a; q) = (aq
; q)
1
where the principal value of q is taken.
If we change q by q 1 we obtain
(0.2.4)
n
(a; q 1)n = (a 1 ; q)n( a)n q ( 2 ) ; a 6= 0:
(0.2.5)
This formula can be used, for instance, to prove the following transformation formula between
the little q-Laguerre (or Wall) polynomials given by (3.20.1) and the q-Laguerre polynomials
dened by (3.21.1) :
pn(x; q jq 1) = (q(q+1; q;)qn) L(n) ( x; q)
n
or equivalently
+1 )n
L(n) (x; q 1) = ((qq; q) ;qqn
pn ( x; qjq):
n
By using (0.2.5) it is not very dicult to verify the following general transformation formula
for 43 polynomials (see section 0.4 for the denition of the basic hypergeometric function 43 ) :
n
q ; a; b; c q 1; q 1 = q n; a 1; b 1; c 1 q; abcqn ;
4 3
4 3
d; e; f d 1; e 1 ; f 1 def
where a limit is needed when one of the parameters is equal to zero. Other transformation formulas
can be obtained from this one by applying limits as discussed in section 0.4.
Finally, we list a number of transformation formulas for the q-shifted factorials, where k and
n are nonnegative integers :
(a; q)n+k = (a; q)n(aqn ; q)k :
(0.2.6)
n
(aq ; q)k = (a; q)k :
(0.2.7)
(aqk ; q)n (a; q)n
(aqk ; q)n k = ((aa;; qq))n ; k = 0; 1; 2; : ::; n:
(0.2.8)
k
n
(a; q)n = (a 1 q1 n; q)n( a)n q( 2 ) ; a 6= 0:
(0.2.9)
n
(aq n; q)n = (a 1 q; q)n( a)n q n ( 2 ) ; a 6= 0:
(0.2.10)
1
n
n
(aq ; q)n = (a q; q)n a ; a 6= 0; b 6= 0:
(0.2.11)
(bq n ; q)n (b 1 q; q)n b
q k k
(2) nk; a 6= 0; k = 0; 1; 2; : ::; n:
(0.2.12)
(a; q)n k = (a 1(aq;1q)nn; q)
a q
k
(a; q)n k = (a; q)n (b 1 q1 n; q)k b k ; a 6= 0; b 6= 0; k = 0; 1; 2; : : :; n:
(0.2.13)
(b; q)n k (b; q)n (a 1 q1 n; q)k a
k
(0.2.14)
(q n ; q)k = (q(;qq; )q)n ( 1)k q(2) nk ; k = 0; 1; 2; : : :; n:
n k
1
(aq n; q)k = (a(a1q1q;kq;)qn) (a; q)k q nk ; a 6= 0:
(0.2.15)
n
n k
1
k
n
(aq n; q)n k = ((aa 1qq;; qq))n aq
q(2) ( 2 ) ; a 6= 0; k = 0; 1; 2; : : :; n:
(0.2.16)
k
(a; q)2n = (a; q2)n (aq; q2)n :
(0.2.17)
9
(a2 ; q2)n = (a; q)n( a; q)n :
(0.2.18)
2
2
(a; q)1 = (a; q )1 (aq; q )1 :
(0.2.19)
(a2 ; q2)1 = (a; q)1 ( a; q)1 :
(0.2.20)
Formula (0.2.18) can be used, for instance, to show that the generating function (3.10.29)
for the continuous q-Legendre polynomials is a q-analogue of the generating function (1.8.44)
for the Legendre polynomials. In fact we obtain (see section 0.4 for the denition of the basic
hypergeometric functions r s )
!
q 21 ei ; q 21 ei q; e i t q 12 e i ; q 21 e
2 1
2 1
q
q
2i 2
i
= 10 qe q2 ; e i t 10 qe q2 ; ei t :
!
i i
q; e t
Now we use the q-binomial theorem (0.5.2) to show that this equals
(qei t; q2)1 (qe i t; q2)1 = q q2 ; ei t q q2 ; e i t :
1 0
(ei t; q2)1 (e i t; q2)1 1 0 If we let q tend to one we now nd by using the binomial theorem (0.5.1)
1
F0
1
2
ei t
1
F0
1
2
e i t = (1 ei t) 21 (1 e i t)
12
; x = cos ;
=p 1
1 2xt + t2
which equals (1.8.44).
0.3 The q-gamma function and the q-binomial coecient
The q-gamma function is dened by
(q; q)1
1 x
(qx ; q)1 (1 q) :
This is a q-analogue of the well-known gamma function since we have
q (x) :=
(0.3.1)
lim
(x) = (x):
q "1 q
Note that the q-gamma function satises the functional equation
1 qz
q (z + 1) = 1 q q (z ); q (1) = 1;
which is a q-extension of the well-known functional equation
(z + 1) = z (z ); (1) = 1
for the ordinary gamma function. For nonintegral values of z this ordinary gamma function also
satises the relation
(z ) (1 z ) = sinz ;
which can be used to show that
lim (1 q +k ) ( ) ( + 1) = ( 1)k+1 ln q; k = 0; 1; 2; : :::
!k
10
This limit can be used to show that the orthogonality relation (3.27.2) for the Stieltjes-Wigert
polynomials follows from the orthogonality relation (3.21.2) for the q-Laguerre polynomials.
The q-binomial coecient is dened by
hni
(q; q)n
n
(0.3.2)
k q = n k q := (q; q)k (q; q)n k ; k = 0; 1; 2; : ::; n;
where n denotes a nonnegative integer.
This denition can be generalized in the following way. For arbitrary complex we have
hi
(q ; q)k ( 1)k qk (k2) :
:=
(0.3.3)
k q
(q; q)k
Or more general for all complex and we have
(q+1 ; q)1 (q +1 ; q)1 :
:=
q ( + 1)
=
(0.3.4)
(q; q)1 (q+1 ; q)1
q
q ( + 1) q ( + 1)
For instance this implies that
(q+1; q)n = n + :
n q
(q; q)n
Note that
( + 1)
=
=
lim
q "1 q
( + 1) ( + 1) :
For integer values of the parameter we have
= ( )k ( 1)k ; k = 0; 1; 2; : ::
k
k!
and when the parameter is an integer too we may write
n = n! ; k = 0; 1; 2; : ::; n; n = 0; 1; 2; : : ::
k k!(n k)!
This latter formula can be used to show that
2n = 21 n 4n; n = 0; 1; 2; : : ::
n!
n
This can be used to write the generating functions (1.8.29) and (1.8.35) for the Chebyshev polynomials of the rst and the second kind in the following form :
R
and
r
1
n
1 2n
t ; R = p1 2xt + t2
1 (1 + R xt) = X
T
(
x
)
n
2
4
n=0 n
n
1 2n + 2
X
t ; R = p1 2xt + t2
4
p
U
(
x
)
=
n
4
R 2(1 + R xt) n=0 n + 1
respectively.
Finally we remark that
(a; q)n =
n h i
X
k=0
n q(k2) ( a)k :
k q
11
(0.3.5)
0.4 Hypergeometric and basic hypergeometric functions
The hypergeometric series r Fs is dened by
1 (a ; : : :; a ) z k
a1; : : :; ar z := X
1
rk ;
F
r s b ; : : :; b (
b
1
s
1 ; : : :; bs )k k !
k=0
(0.4.1)
where
(a1 ; : : :; ar )k := (a1)k (ar )k :
Of course, the parameters must be such that the denominator factors in the terms of the series
are never zero. When one of the numerator parameters ai equals n where n is a nonnegative
integer this hypergeometric series is a polynomial in z . Otherwise the radius of convergence of
the hypergeometric series is given by
8
1 if r < s + 1
>
>
>
>
<
= > 1 if r = s + 1
>
>
>
:
0 if r > s + 1:
A hypergeometric series of the form (0.4.1) is called balanced (or Saalschutzian) if r = s + 1,
z = 1 and a1 + a2 + : : : + as+1 + 1 = b1 + b2 + : : : + bs.
The basic hypergeometric series (or q-hypergeometric series) r s is dened by
1 (a ; : : :; a ; q)
a1; : : :; ar q; z := X
1
r k ( 1)(1+s
r s b ; : : :; b (
b
1 ; : : :; bs ; q )k
1
s
k=0
r)k q(1+s r)(k2)
zk ;
(q; q)k
(0.4.2)
where
(a1 ; : : :; ar ; q)k := (a1 ; q)k (ar ; q)k :
Again, we assume that the parameters are such that the denominator factors in the terms of the
series are never zero. If one of the numerator parameters ai equals q n where n is a nonnegative
integer this basic hypergeometric series is a polynomial in z . Otherwise the radius of convergence
of the basic hypergeometric series is given by
8
1 if r < s + 1
>
>
>
>
<
= > 1 if r = s + 1
>
>
>
:
0
if r > s + 1:
The special case r = s + 1 reads
s+1s
1 (a ; : : :; a ; q) z k
a1 ; : : :; as+1 q; z = X
1
s+1 k
:
(
b
b1; : : :; bs 1 ; : : :; bs ; q )k (q ; q )k
k=0
This basic hypergeometric series was rst introduced by Heine in 1846. Therefore it is sometimes called Heine's series. A basic hypergeometric series of this form is called balanced (or
Saalschutzian) if z = q and a1a2 as+1 q = b1 b2 bs.
The q-hypergeometric series is a q-analogue of the hypergeometric series dened by (0.4.1)
since
a
q 1 ; : : :; qar q; (q 1)1+s r z = F a1; : : :; ar z :
lim
r s b ; : : :; b q"1 r s qb1 ; : : :; qbs 1
s
This limit will be used frequently in chapter 5.
12
We remark that
z
a
a
1 ; : : :; ar 1 ; : : :; ar 1 q;
lim = r 1 s b ; : : :; b q; z :
ar !1 r s b1; : : :; bs ar
1
s
k
In fact, this is the reason for the factors ( 1)(1+s r)k q(1+s r)(2) in the denition (0.4.2) of the
basic hypergeometric series.
The limit relations between hypergeometric orthogonal polynomials listed in chapter 2 of this
report are based on the observations that
a
a
1 ; : : :; ar 1 1 ; : : :; ar 1 ; (0.4.3)
r Fs b ; : : :; b ; z = r 1 Fs 1 b ; : : :; b z ;
1
s 1
1
s 1
a1; : : :; ar 1; ar z = F a1; : : :; ar 1 a z ;
lim
F
(0.4.4)
!1 r s
b1; : : :; bs r 1 s b1; : : :; bs r
a
a
1 ; : : :; ar 1 ; : : :; ar
z
z
=
F
(0.4.5)
lim
F
r
s
1
r
s
!1
b1 ; : : :; bs 1 bs
b1; : : :; bs 1; bs
and
a1 ; : : :; ar 1 ; ar z = F a1; : : :; ar 1 ar z :
lim
F
(0.4.6)
r 1 s 1 b ; : : :; b b
!1 r s b1; : : :; bs 1; bs 1
s 1 s
The limit relations between basic hypergeometric orthogonal polynomials described in chapter
4 of this report are based on the observations that
a
a
1 ; : : :; ar 1 1 ; : : :; ar 1 ; (0.4.7)
r s b ; : : :; b ; q; z = r 1 s 1 b ; : : :; b q; z ;
1
s 1
1
s 1
z
a
a
1 ; : : :; ar 1 1 ; : : :; ar 1 ; ar q
;
a
z
(0.4.8)
q
;
=
lim
!1 r s
b1 ; : : :; bs r 1 s b1; : : :; bs r ;
z
a
a
1 ; : : :; ar 1 ; : : :; ar
q; z = r s 1 b ; : : :; b q; b ;
lim (0.4.9)
!1 r s b1; : : :; bs 1; bs 1
s 1
s
and
a1 ; : : :; ar 1; ar q; z = a1; : : :; ar 1 q; ar z :
(0.4.10)
lim
r 1 s 1 b ; : : :; b b
!1 r s b1 ; : : :; bs 1; bs s
1
s 1
Mostly, the left-hand sides of the formulas (0.4.3) and (0.4.7) occur as limit cases when some
numerator parameter and some denominator parameter tend to the same value.
Finally, we introduce a notation for the N th partial sum of a (basic) hypergeometric series.
We will use this notation in the denitions of the discrete orthogonal polynomials. We also use it
in order to write generating functions for discrete orthogonal polynomials in a compact way. We
dene
N (a ; : : :; a ) z k
X
a
1
rk
1 ; : : :; ar ~
(0.4.11)
r Fs b ; : : :; b z :=
(b ; : : :; b ) k! ;
1
s
k=0
1
sk
where N denotes the nonnegative integer that appears in each denition of a family of discrete
orthogonal polynomials. We also dene
N (a ; : : :; a ; q)
X
a1 ; : : :; ar 1
r k ( 1)(1+s r)k q(1+s r)(k2) z k :
(0.4.12)
r ~s b ; : : :; b q; z :=
(q; q)k
1
s
k=0 (b1 ; : : :; bs; q)k
As an example of the use of these notations we remark that the denition (3.14.1) of the quantum
q-Krawtchouk polynomials must be understood as
n
N (q n; q) (q x ; q)
x X
q
;
q
k
k pqn+1 k :
n
+1
~
q
;
pq
=
2 1
N
N
q
k=0 (q ; q)k (q; q)k
13
In cases of discrete orthogonal polynomials, like the Racah, Hahn, dual Hahn and Krawtchouk
polynomials, we need another special notation for the generating functions. In order to simplify
the notation we write the generating functions as (products of) (truncated as above) power series
in t for which the N th partial sum equals the right-hand side. In these cases we use the notation
' instead of the = sign. As an example of this notation and the one mentioned above we note
that the generating function (1.6.8) of the dual Hahn polynomials must be understood as follows.
The N th partial sum of
et 2F~2
N ( x) (x + + + 1)
x; x + + + 1 t = et X
k
k ( t)k
(
+
1)
(
N
)
k
!
+ 1; N k
k
k=0
equals
N
X
Rn((x); ; ; N ) tn :
n!
n=0
0.5 The q-binomial theorem and other summation formulas
One of the most important summation formulas for hypergeometric series is given by the
binomial theorem :
1 (a)
a z = X
n n
a
F
(0.5.1)
1 0
n! z = (1 z ) ; jz j < 1:
n=0
A q-analogue of this formula is called the q-binomial theorem :
1 (a; q)
a q; z = X
n n (az ; q)1
1 0
(q; q) z = (z ; q) ; jz j < 1:
n=0
n
1
For a = q n with n a nonnegative integer we nd
n
q q; z = (zq n ; q) ; n = 0; 1; 2; : : ::
n
1 0
(0.5.2)
(0.5.3)
In fact this is a q-analogue of Newton's binomium
1 F0
n ( n)
n n z = X
k z k = X n ( z )k = (1 z )n ; n = 0; 1; 2; : : ::
k=0 k!
k=0 k
(0.5.4)
As an example of the use of these formulas we note that the generating function (1.6.5) of the
dual Hahn polynomials can also be written as :
x
N;
x
+
+
+
1
x
~
t :
1 F0
t 2 F1
N
In a similar way we nd for the generating function (3.14.5) of the quantum q-Krawtchouk
polynomials :
(q x t; q)1 qx N ; 0 q; q xt = q x q; t qx N ; 0 q; q xt :
2 1
1 0
pq pq (t; q)1 2 1
Another example of the use of the q-binomial theorem is the proof of the fact that the generating
function (3.10.19) for the continuous q-ultraspherical (or Rogers) polynomials is a q-analogue of
14
the generating function (1.8.14) for the Gegenbauer (or ultraspherical) polynomials. In fact we
have, after the substitution = q :
i
(
; )1 2
( i ; )
(q ei t; qe i t; q)1 = q q; ei t q q; e i t ;
qe tq
=
1 0
1 0
e tq1
(ei t; e i t; q)1
which tends to (for q " 1)
F ei t F e i t = (1 ei t) (1 e i t) = (1 2xt + t2 ) ; x = cos ;
1
0
1
0
which equals (1.8.14).
The well-known Gauss' summation formula
(c) (c a b)
a;
b
2 F1
c 1 = (c a) (c b) ; Re(c a b) > 0
and Vandermonde's summation formula
(c b)n
n;
b
2 F1
c 1 = (c)n ; n = 0; 1; 2; : ::
have the following q-analogues :
a; b q; c = (a 1 c; b 1c; q)1 ; c < 1;
2 1
c ab
(c; a 1b 1c; q)1 ab
n q ; b q; cqn = (b 1c; q)n ; n = 0; 1; 2; : ::
2 1
(c; q)n
c b
and
n q ; b q; q = (b 1 c; q)n bn; n = 0; 1; 2; : : ::
2 1
c (c; q)
n
(0.5.5)
(0.5.6)
(0.5.7)
(0.5.8)
(0.5.9)
On the next level we have the summation formula
(c a)n(c b)n
n; a; b
F
(0.5.10)
3 2
c; 1 + a + b c n 1 = (c)n (c a b)n ; n = 0; 1; 2; : : :
which is called Saalschutz (or Pfa-Saalschutz) summation formula. A q-analogue of this summation formula is
n; a; b (a 1 c; b 1c; q)n
q
(0.5.11)
3 2
c; abc 1q1 n q; q = (c; a 1b 1 c; q)n ; n = 0; 1; 2; : : ::
Finally, we have a summation formula for the 1 1 series :
a c (a 1 c; q)1 :
=
(0.5.12)
q;
1 1
c a
(c; q)1
As an example of the use of this latter formula we remark that the q-Laguerre polynomials
dened by (3.21.1) have the property that
L(n) ( 1; q) = (q;1q) ; n = 0; 1; 2; : : ::
n
15
0.6 Transformation formulas
In this section we list a number of transformation formulas which can be used to transform
denitions or other formulas into equivalent but dierent forms.
First of all we have Heine's transformation formulas for the 2 1 series :
1
b
c;
z
(
az;
b
;
q
)
a;
b
1
q; b
(0.6.1)
q
;
z
=
2
1
2 1
az
c (c; z ; q)1
1
1
= (b (c;c;z ;bzq); q)1 2 1 abc bz z; b q; bc
(0.6.2)
1
1
1
1 = (abc(z ; qz); q)1 2 1 a c;c b c q; abz
(0.6.3)
c :
1
The latter formula is a q-analogue of Euler's transformation formula :
c
a;
c
b
a;
b
c
a
b
z :
(0.6.4)
z
=
(1
z
)
2 F1
2 F1
c
c
Another transformation formula for the 2 F1 series which is also due to Euler is :
a; b z = (1 z ) b F c a; b z :
(0.6.5)
F
2 1
2 1
c z 1
c This transformation formula is also known as the Pfa-Kummer transformation formula.
As a limit case of this one we have Kummer's transformation formula for the conuent hypergeometric series :
a c
a
z
(0.6.6)
1 F1
c z = e 1 F1 c z :
Limit cases of Heine's transformation formulas are
z
1
0
;
0
q
;
z
=
q
;
c
2 1
1
1
c (c; z ; q)1 0 = (z ; 1q) 01 c q; cz ;
1
2
1
a; 0 q; z
c z = ((c;azz;;qq))1 11 az
q; c
1 1 1
a
c
= (z ; q) 1 1 c q; az ;
1
a
(a; z ; q)1 a 1c; 0 q; a
q
;
z
=
1 1
z (c; q)1 2 1
c
1
= (ac 1 z ; q)1 21 a cc; 0 q; azc
and
2 1
a; b q; z = (az; b; q)1 z; 0 q; b
(z ; q)1 2 1 az 0
= ((bzz ;;qq))1 1 1 bzb q; az :
1
16
(0.6.7)
(0.6.8)
(0.6.9)
(0.6.10)
(0.6.11)
(0.6.12)
(0.6.13)
(0.6.14)
If we reverse the order of summation in a terminating 1F1 series we obtain a 2 F0 series, in fact
we have
n
1
(
x
)
n;
a
n
+
1
n
(0.6.15)
1 F1
x ; n = 0; 1; 2; : :::
a x = (a)n 2F0
If we apply this technique to a terminating 2 F1 series we nd
n; b x = (b)n ( x)n F n; c n + 1 1 ; n = 0; 1; 2; : :::
F
(0.6.16)
2 1
2 1
c b n+1 x
(c)n
The q-analogues of these formulas are
n
n
1
n
n+1 1 1 n
(
q
z
)
aq
q
q
;
a
q
; n = 0; 1; 2; : : :
(0.6.17)
1 1
q; z
a q; z = (a; q)n 21
0
and
n q
;
b
2 1
c q; z
n
n+1 1 1 n
n)
cq
q
;
c
q
(
b
;
q
)
n
n
n
(
2
= (c; q) q
( z ) 21
b 1q1 n q; bz ; n = 0; 1; 2; : : :: (0.6.18)
n
A limit case of the latter formula is
n n; 0 q q
;
b
q
n
n
= (b; q)nz 2 1 b 1q1 n q; bz ; n = 0; 1; 2; : : ::
2 0
q; zq
The next transformation formula is due to Jackson :
n n
1
n z ; q )1
1
(
bc
q
q
;
b
q
;
b
c;
0
q; q ; n = 0; 1; 2; : : ::
q
;
z
=
2 1
3
2
c c; b 1cqz 1 (bc 1z ; q)1
Equivalently we have
n
q ; a; 0 q; q = (b 1 q; q)1 q n ; a 1c q; aq ; n = 0; 1; 2; : :::
3 2
c b
b; c (b 1q1 n; q)1 2 1
Other transformation formulas of this kind are given by :
n q ; b q; z 2 1
c n
n
1
1
1 1 n
(
b
c
;
q
)
q
;
qz
;
c
q
bz
n
q; q
= (c; q)
3 2
bc 1q 1 n; 0 q
n
1
n
1
q nz q; q ; n = 0; 1; 2; : : :;
= (b(c; cq;)q)n 32 q bc; b;1 bc
1
n
q ;0 n
or equivalently
n
n
1 (
b
;
q
)
q
;
a;
b
q
;
b
c
q
n
n
3 2
c; 0 q; q = (c; q)n a 21 b 1q1 n q; a
1
n; a bq (
a
c
;
q
)
q
n
n
= (c; q) a 2 1 ac 1q1 n q; c ; n = 0; 1; 2; : :::
n
Limit cases of these formulas are
n
n n q
;
b;
bzq
q
;
b
n
2 0
q; z = b 32
0; 0 q; q ; n = 0; 1; 2; : : :;
17
(0.6.19)
(0.6.20)
(0.6.21)
(0.6.22)
(0.6.23)
(0.6.24)
(0.6.25)
(0.6.26)
or equivalently
3 2
q n ; a; b q; q
0; 0 q n; 0 q; q
=
b 1q1 n a
n n
= an2 0 q ; a q; bqa ; n = 0; 1; 2; : :::
(b; q)nan2 1
(0.6.27)
(0.6.28)
On the next level we have Sears' transformation formula for a terminating balanced 43 series :
n
q
;
a;
b;
c
4 3
d; e; f q; q 1
1
n ; a; b 1d; c 1d (
a
e;
a
f
;
q
)
q
n
n
=
(0.6.29)
(e; f ; q)n a 4 3 d; ae 1q1 n; af 1 q1 n q; q
1
1
a 1 c 1ef ; q)n = (a;(ae; f;ba ef;
1 b 1 c 1 ef ; q )
n
n
1
1
1
1
1
q
;
a
e;
a
f;
a
b
c
ef
q; q ; def = abcq1 n:
(0.6.30)
4 3
a 1 b 1ef; a 1 c 1ef; a 1 q1 n Sears' transformation formula is a q-analogue of Whipple's transformation formula for a terminating balanced 4 F3 series :
n;
a;
b;
c
1 = (e a)n (f a)n 4 F3
d;
e;
f
(e)n (f )n n; a; d b; d c
4 F3
d; a e n + 1; a f n + 1 1 ; a + b + c + 1 = d + e + f + n: (0.6.31)
Whipple's formula can be used to show that the Wilson polynomials dened by (1.1.1) are
symmetric in their parameters in the sense that the following 24 dierent forms are all equal :
Wn (x2 ; a; b; c; d) = Wn (x2 ; a; b; d; c) = Wn (x2 ; a; c; b; d) = = Wn (x2; d; c; b; a):
Sears' transformation formula can be used to derive similar symmetry relations for the AskeyWilson polynomials dened by (3.1.1) :
pn (x; a; b; c; d) = pn (x; a; b; d; c) = pn (x; a; c; b; d) = = pn(x; d; c; b; a):
Finally, we mention a quadratic transformation formula which is due to Singh :
2
2
2 2 2
2
a
;
b
;
c;
d
a
;
b
;
c
;
d
2
2
(0.6.32)
4 3
abq 21 ; abq 21 ; cd q; q = 43 a2 b2q; cd; cdq q ; q ;
which is valid when both sides terminate.
If we apply Singh's formula (0.6.32) to the continuous q-Jacobi polynomials dened by (3.10.1)
and (3.10.2) and use Sears' transformation formula (0.6.29), formula (0.2.10) twice and also formula
(0.2.18), then we nd the quadratic transformation
q)n qn P (;)(x; q):
Pn(;)(xjq2) = ( q(+q;+1
n
; q)n
0.7 Some special functions and their q-analogues
The classical exponential function exp(z ) and the trigonometric functions sin(z ) and cos(z )
can be expressed in terms of hypergeometric functions as
exp(z ) = ez
= 0 F0
18
z ;
(0.7.1)
sin(z ) = z 0 F1
and
3
2
z2 4
(0.7.2)
2
cos(z ) = 0 F1 1 z4 :
(0.7.3)
2
Further we have the well-known Bessel function J (z ) which can be dened by
z 2
2
(0.7.4)
J (z ) := ( + 1) 0F1 + 1 z4 :
Applying this formula to the generating function (1.11.6) of the Laguerre polynomials we
obtain :
1 ()
X
p
(xt) 2 et J(2 xt) = (1+ 1) (Ln+ (1)x) tn:
n
n=0
These functions all have several q-analogues. The exponential function for instance has two
dierent natural q-extensions, denoted by eq (z ) and Eq (z ) dened by
1
n
X
eq (z ) := 1 0 0 q; z = (qz; q)
n=0
and
n
1
X
q(n2 ) z n :
n=0 (q; q)n
These q-analogues of the exponential function are related by
eq (z )Eq ( z ) = 1:
They are q-extensions of the exponential function since
lim
e ((1 q)z ) = lim
E ((1 q)z ) = ez :
q "1 q
q "1 q
Eq (z ) := 0 0
q; z =
If we set a = 0 in the q-binomial theorem we nd for the q-exponential functions :
eq (z ) = 10 0 q; z = (z ; 1q) ; jz j < 1:
1
Further we have
Eq (z ) = 0 0 q; z = ( z ; q)1 :
(0.7.5)
(0.7.6)
(0.7.7)
(0.7.8)
For instance, these formulas can be used to obtain other versions of a generating function for
several sets of orthogonal polynomials mentioned in this report.
If we assume that jz j < 1 we may dene
1
n z 2n+1
X
sinq (z ) := eq (iz ) 2ieq ( iz ) = ( (q1)
n=0 ; q)2n+1
and
(0.7.9)
1 ( 1)n z 2n
X
e
(
iz
)
+
e
(
iz
)
q
q
:
(0.7.10)
=
cosq (z ) :=
2
n=0 (q; q)2n
These are q-analogues of the trigonometric functions sin(z ) and cos(z ). On the other hand we
may dene
Sinq (z ) := Eq (iz ) 2iEq ( iz )
(0.7.11)
19
and
Cosq (z ) := Eq (iz ) +2Eq ( iz ) :
Then it is not very dicult to verify that
eq (iz ) = cosq (z ) + i sinq (z ) and Eq (iz ) = Cosq (z ) + i Sinq (z ):
Further we have
8
<
(0.7.12)
sinq (z )Sinq (z ) + cosq (z )Cosq (z ) = 1
:
sinq (z )Cosq (z ) Sinq (z ) cosq (z ) = 0:
The q-analogues of the trigonometric functions can be used to nd dierent forms of formulas
appearing in this report, although we will not use them.
Some q-analogues of the Bessel functions are given by
2
+1 ; q)1 z z
0
;
0
(
q
(1)
J (z ; q) := (q; q)
(0.7.13)
2 1
q +1 q; 4
1 2
and
+1 z 2 +1; q)1 z q
(
q
(2)
:
(0.7.14)
J (z ; q) := (q; q)
2 0 1 q +1 q;
4
1
These q-Bessel functions are connected by
2
J(2) (z ; q) = ( z4 ; q)1 J(1) (z ; q); jz j < 2:
They are q-analogues of the Bessel function since
lim
J (k)((1 q)z ; q) = J (z ); k = 1; 2:
q "1 These q-Bessel functions were introduced by F.H. Jackson in 1905. They are therefore referred
to as Jackson q-Bessel functions. Other q-analogues of the Bessel function are the so-called HahnExton q-Bessel functions.
As an example we remark that the generating function (3.20.5) for the little q-Laguerre (or
Wall) polynomials can also be written as
1 q(n2 )
( t; q)1 (q; q)1 (xt) 2 J (1) (2pxt; q) = X
pn(x; qjq)tn
(q+1 ; q)1
(
q
;
q
)
n
n=0
or as
1 q(n2 )
(q; q)1 (xt) 2 E (t)J (1) (2pxt; q) = X
n
q
(q+1 ; q)
(q; q) pn (x; q jq)t :
1
n=0
n
0.8 The q-derivative and the q-integral
The q-derivative operator Dq is dened by
8
f (z ) f (qz ) ; z 6= 0
>
>
<
Dq f (z ) := > (1 q)z
>
: 0
f (0);
z = 0:
Further we dene
Dq0f := f and Dqn f := Dq Dqn 1f ; n = 1; 2; 3; : : ::
20
(0.8.1)
(0.8.2)
It is not very dicult to see that
lim
D f (z ) = f 0 (z )
q "1 q
if the function f is dierentiable at z .
An easy consequence of this denition is
Dq [f (x)] = (Dq f ) (x)
(0.8.3)
(0.8.4)
for all real or more general
Dqn [f (x)] = n Dqn f (x); n = 0; 1; 2; : : ::
Further we have
Dq [f (x)g(x)] = f (qx)Dq g(x) + g(x)Dq f (x)
(0.8.5)
which is often referred to as the q-product rule. This can be generalized to a q-analogue of Leibniz'
rule :
n hni
X
n k f (qk x) Dk g (x); n = 0; 1; 2; : :::
(0.8.6)
D
Dqn [f (x)g(x)] =
q
q
k=0 k q
As an example we note that the q-dierence equation (3.21.5) of the q-Laguerre polynomials
can also be written in terms of this q-derivative operator as
(1 q)2xDq2 y(x)+(1 q) 1 q+1 q+2x (Dq y) (qx)+(1 qn)q+1y(qx) = 0; y(x) = L(n)(x; q):
The q-integral is dened by
Z
z
0
1
X
f (t)dq t := z (1 q)
n=0
f (zqn )qn :
(0.8.7)
This denition is due to J. Thomae and F.H. Jackson. Jackson also dened a q-integral on (0; 1)
by
Z 1
1
X
f (qn )qn :
(0.8.8)
f (t)dq t := (1 q)
0
If the function f is continuous on [0; z ] we have
lim
q "1
Z
0
z
f (t)dq t =
n= 1
Z
0
z
f (t)dt:
For instance, the orthogonality relation (3.12.2) for the little q-Jacobi polynomials can also be
written in terms of a q-integral as :
Z1
(qx; q)1 x p (x; q; q jq)p (x; q; q jq)d x
n
q
(q+1 x; q)1 m
0
q++2 ; q)1 (1 q++1 )
(q; q+1 ; q)n qn(+1) ; > 1; > 1:
= (1 q) ((qq;+1
mn
; q+1 ; q)1 (1 q2n+++1 ) (q+1 ; q++1 ; q)n
21
ASKEY-SCHEME
OF
HYPERGEOMETRIC
ORTHOGONAL POLYNOMIALS
1
4
F3 (4)
3
F2 (3)
Continuous
dual Hahn
Continuous
Hahn
Hahn
Dual Hahn
2
F1 (2)
Meixner
Pollaczek
Jacobi
Meixner
Krawtchouk
F1 (1)=2F0(1)
2
F0 (0)
Wilson
Racah
Laguerre
Charlier
Hermite
22
Chapter 1
Hypergeometric orthogonal
polynomials
1.1 Wilson
Denition.
Wn (x2 ; a; b; c; d) = F n; n + a + b + c + d 1; a + ix; a ix 1 :
(1.1.1)
(a + b)n(a + c)n (a + d)n 4 3
a + b; a + c; a + d
Orthogonality. When Re(a; b; c; d) > 0 and non-real parameters occur in conjugate pairs,
then
1
1Z
2
(a + ix) (b + ix) (c + ix) (d + ix) 2 W (x2; a; b; c; d)W (x2 ; a; b; c; d)dx
m
n
(2ix)
0
(n + c + d) ;
= (n + a + b + c + d 1)n n! (n +(2an++ba) +
(1.1.2)
b + c + d) mn
where
(n + a + b) (n + c + d)
(n + a + b) (n + a + c) (n + a + d) (n + b + c) (n + b + d) (n + c + d):
=
If a < 0 and a + b, a + c, a + d are positive or a pair of complex conjugates occur with positive
real parts, then
1
1 Z (a + ix) (b + ix) (c + ix) (d + ix) 2 W (x2 ; a; b; c; d)W (x2; a; b; c; d)dx +
m
n
2 (2ix)
0
+ (a + b) (a + c) (a + (d) 2(ab) a) (c a) (d a) X
(2a)k (a + 1)k (a + b)k (a + c)k (a + d)k
2
2
(k!)(a)k (a b + 1)k (a c + 1)k (a d + 1)k Wm ( (a + k) )Wn ( (a + k) )
k=0;1;2:::
a+k<0
d)
= (n + a + b + c + d 1)nn! (n +(2an++ba) + b +(nc ++ cd+
) mn ;
23
(1.1.3)
where
Wm ( (a + k)2 )Wn ( (a + k)2 ) = Wm ( (a + k)2 ; a; b; c; d)Wn( (a + k)2 ; a; b; c; d):
Recurrence relation.
a2 + x2 W~ n (x2) = AnW~ n+1 (x2 ) (An + Cn) W~ n (x2 ) + CnW~ n 1(x2 );
where
2
W~ n (x2) := W~ n (x2 ; a; b; c; d) = (a +Wb)n((xa +; a;c)b; (c;ad+) d)
n
and
n
(1.1.4)
n
8
>
>
>
>
<
n + a + b)(n + a + c)(n + a + d)
An = (n + a(2+nb++ac++ bd+ c1)(
+ d 1)(2n + a + b + c + d)
>
>
>
>
:
n + b + d 1)(n + c + d 1)
Cn = (2nn(n++ab++bc+ c1)(
+ d 2)(2n + a + b + c + d 1) :
Dierence equation.
n(n + a + b + c + d 1)y(x) = B (x)y(x + i) [B (x) + D(x)] y(x) + D(x)y(x i);
where
y(x) = Wn (x2; a; b; c; d)
and
8
)(c ix)(d ix)
>
>
B (x) = (a ix)(b2ixix
>
>
<
(2ix 1)
>
>
(a + ix)(b + ix)(c + ix)(d + ix) :
>
>
: D(x) =
2ix(2ix + 1)
(1.1.5)
Generating functions.
1 W (x2 ; a; b; c; d)tn
a + ix; b + ix t F c ix; d ix t = X
n
F
:
2 1
a+b 2 1
c+d (
a
n=0 + b)n (c + d)nn!
(1.1.6)
1 W (x2 ; a; b; c; d)tn
a + ix; c + ix t F b ix; d ix t = X
n
F
:
2 1
b+d a+c 2 1
(
a
n=0 + c)n (b + d)nn!
(1.1.7)
1 W (x2 ; a; b; c; d)tn
a + ix; d + ix t F b ix; c ix t = X
n
:
F
2 1
(
a
b+c a+d 2 1
n=0 + d)n (b + c)n n!
(1.1.8)
1
4
t
(
a
+
b
+
c
+
d
1)
;
(
a
+
b
+
c
+
d
)
;
a
+
ix;
a
ix
2
(1 t)
(1 t)2
a + b; a + c; a + d
1
X
b + c + d 1)n W (x2; a; b; c; d)tn:
(1.1.9)
= (a +(ab+
)n (a + c)n(a + d)nn! n
n=0
1
a b c d 4 F3
Remark. If we set
and
1
2
a = 12 ( + + 1) ; b = 21 (2 + 1)
c = 12 (2 + + 1) ; d = 12 ( + 1)
ix ! x + 21 ( + + 1)
24
in
2
W~ n (x2; a; b; c; d) = (a +Wb)n ((xa +; a;c)b; (c;ad+) d) ;
n
dened by (1.1.1) and take
n
n
+ 1 = N or + + 1 = N or + 1 = N; with N a nonnegative integer
we obtain the Racah polynomials dened by (1.2.1).
References. [31], [44], [45], [132], [133], [155], [156], [171], [175], [227], [228].
1.2 Racah
Denition.
Rn((x); ; ; ; ) = 4F~3
where
and
n; n + + + 1; x; x + + + 1 1 ; n = 0; 1; 2; : ::; N; (1.2.1)
+ 1; + + 1; + 1
(x) = x(x + + + 1)
+ 1 = N or + + 1 = N or + 1 = N; with N a nonnegative integer.
Orthogonality.
N
X
( + + 1)x(( + + 3)=2)x( + 1)x ( + + 1)x ( + 1)x R ((x))R ((x))
m
n
x=0 (x!)(( + + 1)=2)x ( + + 1)x ( + 1)x( + 1)x
n( + 1)n( + 1)n( + + 1)nn! ;
= M (n + (+++ 1)
(1.2.2)
mn
+ 2)2n( + 1)n ( + + 1)n ( + 1)n
where
Rn((x)) := Rn((x); ; ; ; )
and
8
>
>
>
>
>
>
>
>
>
<
( + + 2)N ( )N
( + 1)N ( + 1)N
2)N ( )N
M = > ((+ + + +
1)N ( + 1)N
>
>
>
>
>
>
>
( )N ( + + 2)N
>
:
( + 1)N ( + 1)N
if + 1 = N
if + + 1 = N
if + 1 = N:
Recurrence relation.
(x)Rn((x)) = An Rn+1((x)) (An + Cn) Rn((x)) + Cn Rn 1((x));
where
and
Rn((x)) := Rn((x); ; ; ; )
8
>
>
>
>
<
n + + 1)(n + + + 1)(n + + 1)
An = (n + + (2+n 1)(
+ + + 1)(2n + + + 2)
>
>
>
>
:
+ )(n + ) ;
Cn = n(n(2+n+)(n ++ )(2
n + + + 1)
25
(1.2.3)
hence
8
>
>
>
>
>
>
>
>
>
<
(n + N )(n N )(n + + + 1)(n + + 1)
(2n + N )(2n + N + 1)
n + + 1)(n N )(n + + 1)
An = > (n + (2+n++1)(
+ + 1)(2n + + + 2)
>
>
>
>
>
>
>
>
:
and
if + 1 = N
if + + 1 = N
(n + + + 1)(n + + 1)(n + + + 1)(n N ) if + 1 = N
(2n + + + 1)(2n + + + 2)
8
>
>
>
>
>
>
>
>
>
<
Cn = >
>
>
>
>
>
>
>
>
:
n(n + )(n + N 1)(n N 1) if + 1 = N
(2n + N 1)(2n + N )
n(n + )(n + + )(n + + + N + 1) if + + 1 = N
(2n + + )(2n + + + 1)
n(n + )(n + + + N + 1)(n + )
if + 1 = N:
(2n + + )(2n + + + 1)
Dierence equation.
n(n + + + 1)y(x) = B (x)y(x + 1) [B (x) + D(x)] y(x) + D(x)y(x 1);
where
and
(1.2.4)
y(x) = Rn((x); ; ; ; )
8
>
>
>
>
<
x + + + 1)(x + + 1)(x + + + 1)
B (x) = (x + + 1)(
(2x + + + 1)(2x + + + 2)
>
>
>
>
:
+ )(x + + ) :
D(x) = x(x(2+x+)(x + )(2
x + + + 1)
Generating functions.
x
+
+
1
;
x
+
+
+
1
x
+
;
x
~
2 F1
t 2 F1
t
+1
+1
'
N
X
( + + 1)n( + 1)n R ((x); ; ; ; )tn:
n
( + 1)nn!
n=0
(1.2.5)
~ x + + + 1; x + + + 1 t 2F1 x + ; x t
2 F1
++1
+1
N
X
n R ((x); ; ; ; )tn:
' ((+ 1)n+(1)+ 1)
n
n
!
n
n=0
(1.2.6)
x + + 1; x + + + 1 t F x + ; x + t
~
F
2 1
2 1
+1
+ +1
N
X
' ((++1)n (++1)+ 1)n!n Rn((x); ; ; ; )tn:
n
n=0
26
(1.2.7)
4t
( + + 1); 21 ( + + 2); x; x + + + 1 (1 t)2
+ 1; + + 1; + 1
N
X
(1.2.8)
' ( + n!+ 1)n Rn((x); ; ; ; )tn :
n=0
(1 t) 1
~
4 F3
1
2
Remark. If we set = a + b 1, = c + d 1, = a + d 1, = a d and x ! a + ix
in the denition (1.2.1) of the Racah polynomials we obtain the Wilson polynomials dened by
(1.1.1) :
Rn(( a + ix); a + b 1; c + d 1; a + d 1; a d)
2
= W~ n (x2; a; b; c; d) = (a +Wb)n ((xa +; a;c)b; (c;ad+) d) :
n
n
n
References. [31], [43], [45], [47], [90], [156], [180], [183], [192], [194], [227].
1.3 Continuous dual Hahn
Denition.
Sn (x2; a; b; c) = F n; a + ix; a ix 1 :
(1.3.1)
(a + b)n(a + c)n 3 2
a + b; a + c Orthogonality. When a,b and c are positive except possibly for a pair of complex conjugates
with positive real parts, then
1
1 Z (a + ix) (b + ix) (c + ix) 2 S (x2; a; b; c)S (x2 ; a; b; c)dx
n
m
2 (2ix)
0
= (n + a + b) (n + a + c) (n + b + c)n!mn :
(1.3.2)
If a < 0 and a + b, a + c are positive or a pair of complex conjugates with positive real parts, then
1
1 Z (a + ix) (b + ix) (c + ix) 2 S (x2; a; b; c)S (x2; a; b; c)dx +
n
m
2 (2ix)
0
(
a
+ b) (a + c) (b a) (c a) +
( 2a)
X
(2a)k (a + 1)k (a + b)k (a + c)k
k
2
2
(k!)(a)k (a b + 1)k (a c + 1)k ( 1) Sm ( (a + k) )Sn ( (a + k) )
=
where
k=0;1;2:::
a+k<0
(n + a + b) (n + a + c) (n + b + c)n!mn ;
(1.3.3)
Sm ( (a + k)2)Sn ( (a + k)2 ) = Sm ( (a + k)2 ; a; b; c)Sn( (a + k)2; a; b; c):
Recurrence relation.
a2 + x2 S~n (x2 ) = An S~n+1 (x2) (An + Cn) S~n (x2 ) + CnS~n 1 (x2);
where
2
b; c)
S~n (x2) := S~n (x2 ; a; b; c) = (aS+n (bx) ;(a;a +
c)
n
27
n
(1.3.4)
and
8
<
An = (n + a + b)(n + a + c)
:
Cn = n(n + b + c 1):
Dierence equation.
ny(x) = B (x)y(x + i) [B (x) + D(x)] y(x) + D(x)y(x i); y(x) = Sn (x2; a; b; c);
where
8
)(b ix)(c ix)
>
>
B (x) = (a ix2ix
>
>
<
(2ix 1)
>
>
>
>
:
Generating functions.
(1.3.5)
)(b + ix)(c + ix) :
D(x) = (a + ix2ix
(2ix + 1)
1 S (x2 ; a; b; c)
b + ix t = X
n
(1 t) c+ix 2 F1 a + ix;
tn :
a+b (
a
+
b
)
n
!
n
n=0
(1.3.6)
1 S (x2 ; a; b; c)
c + ix t = X
n
tn :
(1 t) b+ix 2F1 a + ix;
(
a
+
c
)
n
!
a+c n
n=0
1 S (x2; a; b; c)
c + ix t = X
n
(1 t) a+ix 2F1 b + ix;
tn:
b+c (
b
+
c
)
n
!
n
n=0
(1.3.7)
(1.3.8)
1 S (x2 ; a; b; c)
X
n
t =
tn:
et 2 F2 aa++ix;b; aa + ix
(
a
+
b
)
(
a
+
c
)
n
!
c n
n
n=0
References. [45], [130], [155], [156], [168], [169].
(1.3.9)
1.4 Continuous Hahn
Denition.
pn(x; a; b; c; d) = in (a + c)n (a + d)n 3 F2
Orthogonality.
n!
n; n + a + b + c + d 1; a + ix 1 :
a + c; a + d
(1.4.1)
1
1 Z (a + ix) (b + ix) (c ix) (d ix)p (x; a; b; c; d)p (x; a; b; c; d)dx
m
n
2
1
) (n + a + d) (n + b + c) (n + b + d) ;
= (2(nn++aa++bc+
(1.4.2)
c + d 1) (n + a + b + c + d 1)n! mn
where
Recurrence relation.
Re(a; b; c; d) > 0; c = a and d = b:
(a + ix)~pn (x) = An p~n+1(x) (An + Cn) p~n (x) + Cn p~n 1(x);
where
p~n(x) := p~n (x; a; b; c; d) = in (a + c)n!(a + d) pn(x; a; b; c; d)
n
28
n
(1.4.3)
and
8
>
>
>
>
<
>
>
>
>
:
n + a + b + c + d 1)(n + a + c)(n + a + d)
An = ((2
n + a + b + c + d 1)(2n + a + b + c + d)
1)(n + b + d 1)
Cn = (2n + a +n(bn++cb++dc 2)(2
n + a + b + c + d 1) :
Dierence equation.
n(n + a + b + c + d 1)y(x) = B (x)y(x + i) [B (x) + D(x)] y(x) + D(x)y(x i);
where
y(x) = pn(x; a; b; c; d)
and
8
< B (x) = (c ix)(d ix)
:
Generating functions.
2
F0
D(x) = (a + ix)(b + ix):
1
a + ix; b + ix it F c ix; d ix it X
pn(x; a; b; c; d)tn:
2
0
n=0
1
X
F1 aa++ixc it 1F1 db + ixd it = (apn+(xc;)a;(bb;+c;dd)) tn :
n
n
n=0
1
1 p (x; a; b; c; d)
a + ix it F c ix it = X
n
F
tn :
1 1
1 1
b+c a+d (
a
+
d
)
(
b
+
c
)
n
n
n=0
(1.4.4)
(1.4.5)
(1.4.6)
(1.4.7)
4t
(a + b + c + d 1); 12 (a + b + c + d); a + ix (1 t)
a + c; a + d
(1 t)2
1
X
= ((aa++bc+) c(a++dd) 1)inn pn(x; a; b; c; d)tn:
(1.4.8)
n
n
n=0
1
a b c d 3 F2
1
2
Remark. Since the generating function (1.4.5) is divergent this relation must be seen as an
equality in terms of formal power series.
References. [29], [31], [46], [51], [120], [156].
1.5 Hahn
Denition.
Qn(x; ; ; N ) = 3 F~2
Orthogonality.
N
X
n; n + + + 1; x 1 ; n = 0; 1; 2; : : :; N:
+ 1; N
+ xN + xQ (x; ; ; N )Q (x; ; ; N )
m
n
x
N x
x=0
1)n n!( + 1)n(n + + + 1)N +1 :
= ((N !)(2
n + + + 1)( N )n( + 1)n mn
(1.5.1)
29
(1.5.2)
Recurrence relation.
xQn(x) = An Qn+1(x) (An + Cn) Qn(x) + CnQn 1(x);
where
Qn(x) := Qn(x; ; ; N )
and
8
>
>
>
>
<
n + + + 1)(n + + 1)(N n)
An = ((2
n + + + 1)(2n + + + 2)
>
>
>
>
:
n(n + )(n + + + N + 1) :
Cn = (2
n + + )(2n + + + 1)
Dierence equation.
where
(1.5.3)
n(n + + + 1)y(x) = B (x)y(x + 1) [B (x) + D(x)] y(x) + D(x)y(x 1);
(1.5.4)
y(x) = Qn (x; ; ; N )
and
Generating functions.
8
<
B (x) = (x N )(x + + 1)
:
D(x) = x(x N 1):
N
X
F1 x +N1 t 1 F1 +x1 t ' (( +N1))nn! Qn (x; ; ; N )tn:
n
n=0
1
(1.5.5)
N
X
( + 1)n
x
N
N
x
+
+
1
~
t
t
'
F
Qn(x; ; ; N )tn: (1.5.6)
1 F1
1
1
N
+ +N +2
(
+
+
N
+
2)
n
!
n
n=0
4t ( + + 1); 21 ( + + 2); x (1 t)
(1 t)2
+ 1; N
N
X
(1.5.7)
' ( + n!+ 1)n Qn(x; ; ; N )tn:
n=0
Remarks. If we interchange the role of x and n in (1.5.1) we obtain the dual Hahn polynomials
dened by (1.6.1).
Since
Qn (x; ; ; N ) = Rx((n); ; ; N )
we obtain the dual orthogonality relation for the Hahn polynomials from the orthogonality relation
(1.6.2) of the dual Hahn polynomials :
N (N !)( N ) ( + 1) (2n + + + 1)
X
n
n
n n!( + 1)n (n + + + 1)N +1 Qn (x; ; ; N )Qn(y; ; ; N )
(
1)
n=0
xy
= + xN
+ x ; x; y 2 f0; 1; 2; : : :; N g:
N x
x
For x = 0; 1; 2; : ::; N the generating function (1.5.5) can also be written as :
X
N ( N)
x
N
x
n
n
1 F1
+ 1 t 1 F1 + 1 t = n=0 ( + 1)n n! Qn (x; ; ; N )t :
References. [10], [25], [27], [31], [45], [47], [77], [80], [82], [87], [89], [104], [105], [124], [144],
[153], [156], [165], [167], [180], [187], [189], [190], [195], [217], [218], [227].
1
~
3 F2
1
2
30
1.6 Dual Hahn
Denition.
Rn((x); ; ; N ) = 3 F~2
where
n; x; x + + + 1 1 ; n = 0; 1; 2; : : :; N;
+ 1; N
(1.6.1)
(x) = x(x + + + 1):
Orthogonality.
N
X
(N !)( N )x ( + 1)x (2x + + + 1) R ((x); ; ; N )R ((x); ; ; N )
m
n
x
(
x=0 1) (x!)( + 1)x (x + + + 1)N +1
= + nmn
(1.6.2)
N + n :
n
N n
Recurrence relation.
(x)Rn((x)) = An Rn+1((x)) (An + Cn) Rn((x)) + Cn Rn 1((x));
where
(1.6.3)
Rn((x)) := Rn((x); ; ; N )
and
Dierence equation.
8
<
An = (n N )(n + + 1)
:
Cn = n(n N 1):
ny(x) = B (x)y(x + 1) [B (x) + D(x)] y(x) + D(x)y(x 1); y(x) = Rn((x); ; ; N ); (1.6.4)
where
8
(x + + + 1)(x + + 1)(N x)
>
>
B
(
x
)
=
>
>
<
(2x + + + 1)(2x + + + 2)
>
>
>
>
:
x(x + )(x + + + N + 1) :
D(x) = (2
x + + )(2x + + + 1)
Generating functions.
N
X
(1 t)x+ 2F~1 x N; x +N + + 1 t ' ( +n!1)n Rn((x); ; ; N )tn :
n=0
(1 t)N x 2F1
N ( N)
x; x t ' X
n R ((x); ; ; N )tn :
n
+1 n
!
n=0
N
X
n R ((x); ; ; N )tn:
(1 t)x 2F1 x N;x +N + 1 t ' ( ( N)n (N+) 1)
n
n
!
n
n=0
et 2F~2
N R ((x); ; ; N )
x; x + + + 1 t ' X
n
tn :
+ 1; N n
!
n=0
31
(1.6.5)
(1.6.6)
(1.6.7)
(1.6.8)
Remarks. If we interchange the role of x and n in the denition (1.6.1) of the dual Hahn
polynomials we obtain the Hahn polynomials dened by (1.5.1).
Since
Rn((x); ; ; N ) = Qx(n; ; ; N )
we obtain the dual orthogonality relation for the dual Hahn polynomials from the orthogonality
relation (1.5.2) for the Hahn polynomials :
N + nN + n
X
N n Rn((x); ; ; N )Rn((y); ; ; N )
n
n=0
x
x (x + + + 1)N +1 ; x; y 2 f0; 1; 2; : ::; N g:
= ((N1)!)((xN!)() (+ 1)
xy
x + 1)x (2x + + + 1)
For x = 0; 1; 2; : ::; N the generating function (1.6.6) can also be written as :
N ( N)
X
x;
x
n
n
N
x
(1 t) 2F1
+ 1 t = n=0 n! Rn((x); ; ; N )t :
For x = 0; 1; 2; : ::; N the generating function (1.6.7) can also be written as :
N ( N ) ( + 1)
X
x
N;
x
+
+
1
n
n R ((x); ; ; N )tn :
x
t =
(1 t) 2 F1
n
N (
N
)
n
!
n
n=0
References. [45], [47], [144], [153], [156], [180], [194], [217], [227].
1.7 Meixner-Pollaczek
Denition.
Orthogonality.
Pn (x; ) =
( )
1
2
Z1
1
Recurrence relation.
e(2
(2)n ein F
2 1
n!
)x j
n; + ix 1 e
2 2
i
:
(1.7.1)
( + ix)j2 Pm()(x; )Pn()(x; )dx
(n + 2) ; > 0 and 0 < < :
= (2 sin
)2n! mn
(1.7.2)
(n + 1)Pn(+1) (x; ) 2 [x sin + (n + ) cos ] Pn()(x; ) + (n + 2 1)Pn()1(x; ) = 0: (1.7.3)
Dierence equation.
ei ( ix)y(x + i) + 2i [x cos (n + ) sin ] y(x) e i ( + ix)y(x i) = 0;
where
y(x) = Pn() (x; ):
Generating functions.
1
X
(1 ei t) +ix(1 e i t) ix = Pn()(x; )tn:
n=0
(1.7.4)
(1.7.5)
1 P ()(x; )
+ ix (e 2i 1)t = X
n
n
(1.7.6)
in t :
(2
)
e
2 n
n=0
References. [10], [15], [25], [31], [45], [47], [72], [77], [126], [156], [174], [180], [189], [227],
[230].
e t 1 F1
32
1.8 Jacobi
Denition.
Pn;) (x) =
(
Orthogonality.
Z1
( + 1)n F
n! 2 1
n; n + + + 1 1 x :
2
+1
(1.8.1)
(1 x)(1 + x) Pm(;)(x)Pn(;)(x)dx
1
++1
= 2n +2 + + 1 (nn+! (n++1) +(n++1)+ 1) mn ; > 1 and > 1:
(1.8.2)
Recurrence relation.
+ + + 1)
(; )
xPn(;)(x) = (2n +2(n++1)(+n1)(2
n + + + 2) Pn+1 (x) +
2
2
)(n + )
(; )
+ (2n + + )(2n+ + + 2) Pn(;)(x) + (2n + 2(+n +)(2
n + + + 1) Pn 1 (x): (1.8.3)
Dierential equation.
(1 x2)y00 (x)+[ ( + + 2)x] y0 (x)+ n(n + + +1)y(x) = 0; y(x) = Pn(;) (x): (1.8.4)
Generating functions.
1
X
2+
(; )
n ; R = p1 2xt + t2 :
=
P
(
x
)
t
n
R(1 + R t) (1 + R + t) n=0
1
X
(x + 1)t
(x 1)t
Pn(;) (x) tn :
=
F
F
0 1
0 1
+ 1 2
+ 1 2
n=0 ( + 1)n( + 1)n
( + + 1); 21 ( + + 2) 2(x 1)t
(1 t)
2 F1
(1 t)2
+1
1
X
= ((+ + +1) 1)n Pn(;)(x)tn :
n
n=0
1
1
2
( + + 1); 21 ( + + 2) 2(x + 1)t
(1 + t)2
+1
1
X
= ((+ + +1) 1)n Pn(;)(x)tn :
n
n=0
(1 + t) 1
2 F1
1
2
(1.8.5)
(1.8.6)
(1.8.7)
(1.8.8)
1 R+t
1 R t
;
+
+
1
;
+
+
1
2 F1
2 F1
+1
2
2
+1
1
p
X ( )n ( + + 1 )n
=
Pn(;)(x)tn ; R = 1 2xt + t2 ; arbitrary: (1.8.9)
(
+
1)
(
+
1)
n
n
n=0
Remarks. The Jacobi polynomials dened by (1.8.1) and the Meixner polynomials given by
(1.9.1) are related in the following way :
( )n M (x; ; c) = P ( 1; n x) 2 c :
n
n! n
c
33
The Jacobi polynomials are also related to the Gegenbauer (or ultraspherical) polynomials
dened by (1.8.10) by the quadratic transformations :
1 1
C2(n) (x) = (1 ))n Pn( 2 ; 2 ) (2x2 1)
(2 n
and
1 1
C2(n)+1(x) = ((1 ))n+1 xPn( 2 ; 2 ) (2x2 1):
2 n+1
References. [2], [3], [7], [9], [14], [25], [26], [27], [28], [31], [33], [34], [35], [42], [45], [50], [55],
[61], [68], [69], [77], [80], [82], [90], [93], [94], [98], [99], [100], [101], [102], [103], [104], [105], [106],
[115], [119], [122], [123], [124], [134], [146], [147], [148], [149], [151], [152], [155], [156], [165], [170],
[173], [176], [178], [180], [182], [184], [185], [186], [188], [193], [203], [205], [211], [213], [214], [218],
[220], [229], [232].
Special cases
1.8.1 Gegenbauer / Ultraspherical
Denition. The Gegenbauer (or ultraspherical) polynomials are Jacobi polynomials with
==
1
2
and another normalization :
Cn (x) =
( )
(2)n P ( 21 ;
( + 12 )n n
21 ) (x) =
(2)n F
n! 2 1
n; n + 2 1 x ; 6= 0:
+ 21 2
(1.8.10)
Orthogonality.
Z1
1
1 2
mn ; > 21 and 6= 0:
(1 x2 ) 21 Cm()(x)Cn()(x)dx = (n +2 2)2
f ()g (n + )n!
(1.8.11)
Recurrence relation.
2(n + )xCn() (x) = (n + 1)Cn(+1) (x) + (n + 2 1)Cn()1(x):
(1.8.12)
(1 x2)y00 (x) (2 + 1)xy0 (x) + n(n + 2)y(x) = 0; y(x) = Cn()(x):
(1.8.13)
Dierential equation.
Generating functions.
1
X
(1 2xt + t2 ) = Cn()(x)tn :
n=0
R
1
1 + R xt
2
1
2 =
1
X
p
( + 12 )n () n
C
(
x
)
t
;
R
=
1 2xt + t2:
n
(2
)
n
n=0
1
X
(x 1)t
(x + 1)t
Cn()(x) tn :
F
F
=
0 1
0 1
1
+ 12 2
+ 21 2
n=0 (2)n ( + 2 )n
ext 0 F1
1
+ 2
1 C ()(x)
(x2 1)t2 = X
n
tn :
4
(2
)
n
n=0
34
(1.8.14)
(1.8.15)
(1.8.16)
(1.8.17)
2
(1 xt)
F1
2 F1
; 2 1 R t F ; 2 1 R + t 2 1
+ 12 2
+ 12 2
1 ( ) (2 )
X
n
n C () (x)tn; R = p1 2xt + t2; arbitrary: (1.8.18)
= (2
n
1
n=0 )n ( + 2 )n
1
2
!
1 ( )
; 12 + 21 (x2 1)t2 = X
n C ()(x)tn; arbitrary:
1
2
(1
xt
)
(2
)
+ 2
n n
n=0
(1.8.19)
Remarks. The case = 0 needs another normalization. In that case we have the Chebyshev
polynomials of the rst kind described in the next subsection.
The Gegenbauer (or ultraspherical) polynomials dened by (1.8.10) and the Jacobi polynomials
given by (1.8.1) are related by the quadratic transformations :
1 1
C2(n) (x) = (1 ))n Pn( 2 ; 2 ) (2x2 1)
(2 n
and
1 1
C2(n)+1(x) = ((1 ))n+1 xPn( 2 ; 2 ) (2x2 1):
2
n+1
References. [2], [4], [26], [27], [31], [33], [41], [54], [55], [56], [61], [62], [63], [64], [65], [66],
[68], [77], [83], [92], [94], [103], [108], [119], [123], [156], [164], [180], [203], [205], [220], [223], [232].
1.8.2 Chebyshev
Denitions. The Chebyshev polynomials of the rst kind can be obtained from the Jacobi
polynomials by taking = = 21 :
1
1
( 2; 2)
Tn(x) = Pn( 12 ; 21 )(x) = 2 F1 n;1 n 1 2 x
(1.8.20)
2
Pn
(1)
and the Chebyshev polynomials of the second kind can be obtained from the Jacobi polynomials
by taking = = 21 :
1 1
(2;2)
Un (x) = (n + 1) Pn( 12 ; 21 )(x) = (n + 1)2 F1
Pn (1)
Orthogonality.
Z1
(1 x2)
12
Tm (x)Tn (x)dx =
1
Z1
Recurrence relations.
1
8
>
<
>
:
n; n + 2 1 x :
3
2
2
; n 6= 0
2 mn
mn ; n = 0:
(1 x2) 12 Um (x)Un (x)dx = 2 mn :
2xTn (x) = Tn+1 (x) + Tn 1(x); T0 (x) = 1 and T1 (x) = x:
35
(1.8.21)
(1.8.22)
(1.8.23)
(1.8.24)
2xUn(x) = Un+1 (x) + Un 1(x):
(1.8.25)
(1 x2 )y00 (x) xy0 (x) + n2y(x) = 0; y(x) = Tn (x):
(1.8.26)
(1 x2)y00 (x) 3xy0 (x) + n(n + 2)y(x) = 0; y(x) = Un (x):
(1.8.27)
1
1 xt = X
n
1 2xt + t2 n=0 Tn (x)t :
(1.8.28)
Dierential equations.
Generating functions.
R
r
1
0 F1
1 1
1 (1 + R xt) = X
2 n
n ; R = p1 2xt + t2 :
T
(
x
)
t
n
2
n=0 n!
1
2
(x 1)t
2
ext0 F1
1 T (x)
(x + 1)t = X
n tn :
1
1
2
2
n=0 2 n n!
2
1 T (x)
(x
1)t2 = X
n tn:
1 4
2
n=0 n!
0 F1
(1 xt)
1
2
1
3
X
1
2 n
n ; R = p1 2xt + t2 :
q
U
(
x
)
t
=
n
R 12 (1 + R xt) n=0 (n + 1)!
2
(1 xt)
F1
2 F1
3
2
1
(x + 1)t = X
Un (x) tn :
3
3
2
2
n=0 2 n (n + 1)!
X
2
1
2
) tn :
ext 0F1 3 (x 4 1)t = (nUn+(x1)!
2
n=0
(x 1)t
2
0 F1
(1.8.32)
(1.8.33)
(1.8.34)
(1.8.35)
(1.8.36)
(1.8.37)
; 2 1 R t F ; 2 1 R + t 2 1
3
3
2
2
2
2
1 ( ) (2 )
p
X
n
n U (x)tn ; R = 1 2xt + t2 ; arbitrary:
=
n
3
n=0 2 n (n + 1)!
1
2
(1.8.31)
!
1 ( )
X
; 12 + 21 (x2 1)t2
n T (x)tn; arbitrary:
=
n
1
2
(1
xt
)
n
!
2
n=0
1
X
1
n
=
1 2xt + t2 n=0 Un (x)t :
0 F1
(1.8.30)
; 1 R t F ; 1 R + t F
2 1
2 1
1
1
2
2
2
2
1
p
X
= ( )1n( )n Tn (x)tn; R = 1 2xt + t2; arbitrary:
n=0 2 n n!
2 F1
(1.8.29)
(1.8.38)
!
1 ( )
; 12 + 21 (x2 1)t2 = X
n U (x)tn ; arbitrary:
3
2
(1
xt
)
(
n
+
1)! n
2
n=0
36
(1.8.39)
Remarks. The Chebyshev polynomials can also be written as :
Tn (x) = cos(n arccos x)
and
n + 1) ; x = cos :
Un (x) = sin(sin
Further we have
Un (x) = Cn(1)(x)
where Cn() (x) denotes the Gegenbauer (or ultraspherical) polynomial dened by (1.8.10) in the
preceding subsection.
References. [2], [33], [36], [77], [83], [94], [119], [170], [180], [205], [206], [210], [220], [232].
1.8.3 Legendre / Spherical
Denition. The Legendre (or spherical) polynomials are Jacobi polynomials with = = 0 :
;
Pn(x) = Pn (x) = 2 F1
(0 0)
Orthogonality.
Z1
1
Recurrence relation.
n; n + 1 1 x :
1 2
Pm (x)Pn(x)dx = 2n 2+ 1 mn :
(1.8.40)
(1.8.41)
(2n + 1)xPn(x) = (n + 1)Pn+1 (x) + nPn 1(x):
(1.8.42)
Dierential equation.
(1 x2)y00 (x) 2xy0 (x) + n(n + 1)y(x) = 0; y(x) = Pn(x):
Generating functions.
1
X
1
P (x)tn:
=
1 2xt + t2 n=0 n
1 P (x)
X
(x 1)t
(x + 1)t
n tn:
F
F
=
0 1
0 1
2
1 2
1 2
n
n=0 !
p
ext 0 F1 1 (x 4 1)t
2
(1 xt)
F1
2 F1
2
2
=
1
X
Pn(x) tn :
n=0 n!
1
2
(1.8.45)
1 ( )
; 21 + 12 (x2 1)t2 = X
n P (x)tn ; arbitrary:
n
(1 xt)2
n
!
1
n=0
(1.8.44)
(1.8.46)
; 1 1 R t F ; 1 1 R + t
2 1
1 2
1 2
1
p
X
= ( )n (1n!2 )n Pn (x)tn; R = 1 2xt + t2; arbitrary:
n=0
(1.8.43)
(1.8.47)
(1.8.48)
References. [2], [5], [10], [53], [55], [77], [83], [94], [119], [180], [182], [205], [220], [232].
37
1.9 Meixner
Denition.
Mn (x; ; c) = 2F1
Orthogonality.
n; x 1 1 :
c
1
X
( )x cx M (x; ; c)M (x; ; c) = c nn! ; > 0 and 0 < c < 1:
m
n
( )n (1 c) mn
x=0 x!
(1.9.1)
(1.9.2)
Recurrence relation.
(c 1)xMn (x; ; c) = c(n + )Mn+1 (x; ; c) +
[n + (n + )c] Mn(x; ; c) + nMn 1(x; ; c):
(1.9.3)
Dierence equation.
n(c 1)y(x) = c(x + )y(x + 1) [x + (x + )c] y(x) + xy(x 1); y(x) = Mn (x; ; c): (1.9.4)
Generating functions.
1 ct
x
1
X
(1 t) x = (n)!n Mn (x; ; c)tn:
n=0
(1.9.5)
1 M (x; ; c)
x 1 c t = X
n
tn:
(1.9.6)
c
n
!
n=0
Remarks. The Meixner polynomials dened by (1.9.1) and the Jacobi polynomials given by
(1.8.1) are related in the following way :
( )n M (x; ; c) = P ( 1; n x) 2 c :
n
n! n
c
The Meixner polynomials are also related to the Krawtchouk polynomials dened by (1.10.1)
in the following way :
Kn (x; p; N ) = Mn x; N; p p 1 :
e t 1 F1
References. [7], [10], [15], [17], [25], [27], [31], [36], [45], [47], [77], [82], [94], [97], [104], [105],
[129], [135], [143], [156], [174], [180], [189], [222], [233].
1.10 Krawtchouk
Denition.
Orthogonality.
Kn (x; p; N ) = 2F~1
n; x 1 ; n = 0; 1; 2; : : :; N:
N p
N X
x=0
(1.10.1)
N px (1 p)N x K (x; p; N )K (x; p; N ) = ( 1)n n! 1 p n ; 0 < p < 1: (1.10.2)
mn
m
n
( N )n
p
x
38
Recurrence relation.
xKn (x; p; N ) = p(N n)Kn+1 (x; p; N ) +
[p(N n) + n(1 p)] Kn (x; p; N ) + n(1 p)Kn 1(x; p; N ):
Dierence equation.
ny(x) = p(N x)y(x + 1) [p(N x) + x(1 p)] y(x) + x(1 p)y(x 1);
where
(1.10.4)
y(x) = Kn (x; p; N ):
Generating functions.
(1.10.3)
x
N N X
Kn (x; p; N )tn:
1 (1 p p) t (1 + t)N x '
n
n=0
(1.10.5)
N K (x; p; N )
x t ' X
n
n
(1.10.6)
N p n=0
n! t :
Remarks. The Krawtchouk polynomials are self-dual, which means that
Kn (x; p; N ) = Kx (n; p; N ); n; x 2 f0; 1; 2; : ::; N g:
By using this relation we easily obtain the so-called dual orthogonality relation from the orthogonality relation (1.10.2) :
1 p x
N N
X
n(1 p)N n Kn (x; p; N )Kn(y; p; N ) = p xy ;
p
N
n=0 n
x
where 0 < p < 1 and x; y 2 f0; 1; 2; : : :; N g.
The Krawtchouk polynomials are related to the Meixner polynomials dened by (1.9.1) in the
following way :
Kn (x; p; N ) = Mn x; N; p p 1 :
et 1 F~1
For x = 0; 1; 2; : ::; N the generating function (1.10.5) can also be written as :
x
N N X
1 (1 p p) t (1 + t)N x =
Kn (x; p; N )tn:
n
n=0
References. [10], [25], [27], [31], [45], [77], [87], [90], [91], [94], [104], [105], [143], [154], [156],
[167], [180], [189], [191], [217], [218], [220], [233].
1.11 Laguerre
Denition.
Orthogonality.
Ln
( )
Z1
0
(x) = ( +n!1)n 1 F1 +n1 x :
x e x L(m) (x)L(n) (x)dx = (n +n! + 1) mn ; > 1:
39
(1.11.1)
(1.11.2)
Recurrence relation.
(n + 1)L(n+1) (x) (2n + + 1 x)L(n) (x) + (n + )L(n)1 (x) = 0:
(1.11.3)
Dierential equation.
xy00 (x) + ( + 1 x)y0 (x) + ny(x) = 0; y(x) = L(n) (x):
Generating functions.
(1 t)
1
xt
1
X
exp t 1 = L(n) (x)tn:
n=0
1 ()
X
et 0F1 + 1 xt = (Ln+ (1)x) tn:
n
n=0
(1.11.4)
(1.11.5)
(1.11.6)
1 ( )
X
xt
n
()
n
(1.11.7)
=
(1 t)
+ 1 t 1 n=0 ( + 1)n Ln (x)t ; arbitrary:
Remarks. The denition (1.11.1) of the Laguerre polynomials can also be written as :
1 F1
n
X
L(n) (x) = n1! ( kn!)k ( + k + 1)n k xk :
k=0
In this way the Laguerre polynomials can be dened for all . Then we have the following
connection with the Charlier polynomials dened by (1.12.1) :
( a)n C (x; a) = L(x n) (a):
n
n! n
The Laguerre polynomials dened by (1.11.1) and the Hermite polynomials dened by (1.13.1)
are connected by the following quadratic transformations :
1
H2n(x) = ( 1)n n!22nL(n 2 ) (x2)
and
1
H2n+1(x) = ( 1)n n!22n+1xL(n2 ) (x2):
In combinatorics the Laguerre polynomials with = 0 are often called Rook polynomials.
References. [1], [2], [3], [7], [9], [10], [14], [15], [25], [27], [31], [35], [36], [40], [45], [52], [55],
[57], [58], [61], [65], [66], [67], [68], [70], [75], [77], [81], [82], [94], [95], [106], [116], [117], [119],
[123], [124], [129], [134], [135], [139], [141], [143], [150], [152], [155], [156], [174], [180], [182], [205],
[210], [211], [220], [222].
1.12 Charlier
Denition.
Orthogonality.
Cn (x; a) = 2F0
1
X
n; x 1 :
a
ax C (x; a)C (x; a) = n!a nea ; a > 0:
m
n
mn
x=0 x!
40
(1.12.1)
(1.12.2)
Recurrence relation.
xCn(x; a) = aCn+1 (x; a) (n + a)Cn(x; a) + nCn 1(x; a):
(1.12.3)
Dierence equation.
ny(x) = ay(x + 1) (x + a)y(x) + xy(x 1); y(x) = Cn (x; a):
(1.12.4)
Generating function.
x
1
X
et 1 at = Cn(nx!; a) tn:
(1.12.5)
n=0
Remark. The denition (1.11.1) of the Laguerre polynomials can also be written as :
n
X
L(n) (x) = n1! ( kn!)k ( + k + 1)n k xk :
k=0
In this way the Laguerre polynomials can be dened for all . Then we have the following
connection with the Charlier polynomials dened by (1.12.1) :
( a)n C (x; a) = L(x n) (a):
n
n! n
References. [7], [10], [15], [17], [25], [27], [45], [77], [78], [87], [94], [104], [105], [129], [156],
[167], [174], [180], [220], [222], [233].
1.13 Hermite
Denition.
Hn(x) = (2x)n2 F0
Orthogonality.
Z1
n=2; (n 1)=2 1
x2 :
(1.13.1)
x2 Hm (x)Hn(x)dx = 2nn!pmn :
(1.13.2)
Hn+1(x) 2xHn(x) + 2nHn 1(x) = 0:
(1.13.3)
y00 (x) 2xy0 (x) + 2ny(x) = 0; y(x) = Hn(x):
(1.13.4)
Recurrence relation.
1
e
Dierential equation.
Generating functions.
exp 2xt t2 =
1
X
Hn(x) tn :
n=0 n!
8
>
>
>
>
>
<
1
n
p X
et cos(2x t) = ((2n1))! H2n(x)tn
>
>
>
>
>
:
1
n
t
p X
pe sin(2x t) = (2(n +1)1)! H2n+1(x)tn:
t
n=0
n=0
41
(1.13.5)
(1.13.6)
8
>
>
>
>
>
<
>
>
>
>
>
:
e
e
1
t2 cosh(2xt) = X H2n(x) t2n
n=0 (2n)!
t2 sinh(2xt) =
1
X
H2n+1(x) t2n+1 :
n=0 (2n + 1)!
8
>
>
>
>
>
>
<
(1 + t )
>
>
>
>
>
>
:
1 ( + 1 )
1
2 2
X
2 n
p xt 2 1F1 +3 2 1x+tt2 = (2
H2n+1(x)t2n+1:
n
+
1)!
1+t
2
n=0
2
1 F1
1 ( )
x2t2 = X
n H (x)t2n
1
2
1
+
t
(2
n
)! 2n
2
n=0
!
1 H (x)
1 + 2xt + 4t2 exp 4x2t2 = X
n tn;
32
2
2
1
+
4
t
[
n=
2]!
(1 + 4t )
n=0
where [] denotes the largest integer smaller than or equal to .
Remarks. The Hermite polynomials can also be written as :
(1.13.7)
(1.13.8)
(1.13.9)
n=2]
Hn (x) = [X
( 1)k (2x)n 2k ;
n!
k=0 k!(n 2k)!
where [] denotes the largest integer smaller than or equal to .
The Laguerre polynomials dened by (1.11.1) and the Hermite polynomials dened by (1.13.1)
are connected by the following quadratic transformations :
1
H2n(x) = ( 1)n n!22nLn( 2 ) (x2)
and
1
H2n+1(x) = ( 1)n n!22n+1xL(n2 ) (x2):
References. [2], [7], [10], [14], [15], [25], [27], [31], [35], [45], [49], [55], [57], [58], [71], [77],
[81], [83], [94], [118], [119], [123], [156], [174], [180], [182], [205], [210], [220], [222], [225], [231].
42
Chapter 2
Limit relations between
hypergeometric orthogonal
polynomials
2.1 Wilson ! Continuous dual Hahn
The continuous dual Hahn polynomials can be found from the Wilson polynomials dened by
(1.1.1) by dividing by (a + d)n and letting d ! 1 :
Wn (x2 ; a; b; c; d) = S (x2 ; a; b; c);
lim
n
d!1
(a + d)n
where Sn (x2 ; a; b; c) is dened by (1.3.1).
2.2 Wilson ! Continuous Hahn
The continuous Hahn polynomials dened by (1.4.1) are obtained from the Wilson polynomials
by the substitution a ! a it, b ! b it, c ! c + it, d ! d + it and x ! x + t in the denition
(1.1.1) of the Wilson polynomials and the limit t ! 1 in the following way :
W
n (x + t)2 ; a it; b it; c + it; d + it
= pn(x; a; b; c; d):
tlim
!1
( 2t)n n!
2.3 Wilson ! Jacobi
The Jacobi polynomials given by (1.8.1) can be found from the Wilson
polynomials by substituting
q
1
1
1
1
a = b = 2 ( +1), c = 2 ( +1)+ it, d = 2 ( +1) it and x ! t 2 (1 x) in the denition (1.1.1)
of the Wilson polynomials and taking the limit t ! 1. In fact we have
W
n 12 (1 x)t2 ; 12 ( + 1); 21 ( + 1); 21 ( + 1) + it; 21 ( + 1) it
lim
= Pn(;)(x):
t!1
t2nn!
2.4 Racah ! Hahn
If we take +1 = N and let ! 1 in the denition (1.2.1) of the Racah polynomials, we obtain
the Hahn polynomials dened by (1.5.1). Hence
lim R ((x); ; ; N 1; ) = Qn(x; ; ; N ):
!1 n
43
The Hahn polynomials can also be obtained from the Racah polynomials by taking = N 1
in the denition (1.2.1) and letting ! 1 :
lim R ((x); ; ; ; N 1) = Qn (x; ; ; N ):
!1 n
Another way to do this is to take +1 = N and ! + + N +1 in the denition (1.2.1) of the
Racah polynomials and then take the limit ! 1. In that case we obtain the Hahn polynomials
given by (1.5.1) in the following way :
lim R ((x); N 1; + + N + 1; ; ) = Qn(x; ; ; N ):
!1 n
2.5 Racah ! Dual Hahn
If we take + 1 = N and let ! 1 in (1.2.1), then we obtain the dual Hahn polynomials from
the Racah polynomials. So we have
lim R ((x); N 1; ; ; ) = Rn((x); ; ; N ):
!1 n
And if we take = N 1 and let ! 1 in (1.2.1), then we also obtain the dual Hahn
polynomials :
lim R ((x); ; N 1; ; ) = Rn ((x); ; ; N ):
!1 n
Finally, if we take + 1 = N and ! + + N + 1 in the denition (1.2.1) of the Racah
polynomials and take the limit ! 1 we nd the dual Hahn polynomials given by (1.6.1) in the
following way :
lim R ((x); ; ; N 1; + + N + 1) = Rn((x); ; ; N ):
!1 n
2.6 Continuous dual Hahn ! Meixner-Pollaczek
The Meixner-Pollaczek polynomials given by (1.7.1) can be obtained from the continuous dual
Hahn polynomials by the substitutions x ! x t, a = + it, b = it and c = t cot in the
denition (1.3.1) and the limit t ! 1 :
(x t)2; + it;
it;
t
cot
S
n
= Pn()(x; ):
lim
t!1
t
n!
sin n
2.7 Continuous Hahn ! Meixner-Pollaczek
By taking x ! x t, a = + it, c = it and b = d = t tan in the denition (1.4.1) of
the continuous Hahn polynomials and taking the limit t ! 1 we obtain the Meixner-Pollaczek
polynomials dened by (1.7.1) :
pn (x t; + it; t tan
; it; t tan ) = P ()(x; ):
lim
n
t!1
it
n
i
cos n
2.8 Continuous Hahn ! Jacobi
The Jacobi polynomials dened by (1.8.1) follow from the continuous Hahn polynomials by the
substitution x ! 12 xt, a = 12 ( +1+ it), b = 21 ( +1 it), c = 12 ( +1 it) and d = 21 ( +1+ it)
in (1.4.1), division by ( 1)n tn and the limit t ! 1 :
1
xt
; 21 ( + 1 + it); 12 ( + 1 it); 21 ( + 1 it); 12 ( + 1 + it) = P (;)(x):
p
n
2
lim
n
t!1
( 1)n tn
44
2.9 Hahn ! Jacobi
To nd the Jacobi polynomials from the Hahn polynomials we take x ! Nx in (1.5.1) and let
N ! 1: We have
Pn(;)(1 2x) :
lim
Q
(
Nx
;
;
;
N
)
=
n
N !1
Pn(;)(1)
2.10 Hahn ! Meixner
If we take = b 1, = N (1 c)c 1 in the denition (1.5.1) of the Hahn polynomials and let
N ! 1 we nd the Meixner polynomials given by (1.9.1) :
1 c ; N = M (x; b; c):
x
;
b
1
;
N
lim
Q
n
N !1 n
c
2.11 Hahn ! Krawtchouk
If we take = pt and = (1 p)t in the denition (1.5.1) of the Hahn polynomials and let t ! 1
we obtain the Krawtchouk polynomials dened by (1.10.1) :
lim Q (x; pt; (1 p)t; N ) = Kn (x; p; N ):
t!1 n
2.12 Dual Hahn ! Meixner
To obtain the Meixner polynomials from the dual Hahn polynomials we have to take = 1
and = N (1 c)c 1 in the denition (1.6.1) of the dual Hahn polynomials and let N ! 1 :
1
c
lim R (x); 1; N c ; N = Mn (x; ; c):
N !1 n
2.13 Dual Hahn ! Krawtchouk
In the same way we nd the Krawtchouk polynomials from the dual Hahn polynomials by setting
= pt, = (1 p)t in (1.6.1) and let t ! 1 :
lim R ((x); pt; (1 p)t; N ) = Kn (x; p; N ):
t!1 n
2.14 Meixner-Pollaczek ! Laguerre
The Laguerre polynomials can be obtained from the Meixner-Pollaczek polynomials dened by
(1.7.1) by the substitution = 12 ( + 1), x ! 21 1x and letting ! 0 :
1
1
lim P ( 2 + 2 )
!0 n
x ; = L() (x):
n
2
2.15 Meixner-Pollaczek ! Hermite
p
If we substitute x ! (sin ) 1 (x cos ) in the denition (1.7.1) of the Meixner-Pollaczek
polynomials and then let ! 1 we obtain the Hermite polynomials :
lim !1
n
2 Pn
( )
!
p
x cos ; = Hn(x) :
sin n!
45
2.16 Jacobi ! Laguerre
The Laguerre polynomials can be obtained from the Jacobi polynomials dened by (1.8.1) by
letting x ! 1 2 1 x and then ! 1 :
2
x
(; )
lim P
1 = L(n) (x):
!1 n
2.17 Jacobi ! Hermite
The Hermite polynomials given by (1.13.1) follow from the Jacobi polynomials dened by (1.8.1)
by taking = and letting ! 1 in the following way :
Hn(x) :
n (;) x
2 Pn
=
lim
1
!1
2n n!
2
2.18 Meixner ! Laguerre
If we take = + 1 and x ! (1 c) 1 x in the denition (1.9.1) of the Meixner polynomials and
let c ! 1 we obtain the Laguerre polynomials :
lim M
c!1 n
x ; + 1; c = L(n)(x) :
1 c
L(n) (0)
2.19 Meixner ! Charlier
If we take c = (a + ) 1 a in the denition (1.9.1) of the Meixner polynomials and let ! 1 we
nd the Charlier polynomials :
a
lim M x; ; a + = Cn (x; a):
!1 n
2.20 Krawtchouk ! Charlier
The Charlier polynomials given by (1.12.1) can be found from the Krawtchouk polynomials dened
by (1.10.1) by taking p = N 1a and let N ! 1 :
a
;
N
= Cn (x; a):
x
;
lim
K
N !1 n
N
2.21 Krawtchouk ! Hermite
The Hermiteppolynomials follow from the Krawtchouk polynomials dened by (1.10.1) by setting
x ! pN + x 2p(1 p)N and then letting N ! 1 :
lim
N !1
s N K pN + xp2p(1 p)N ; p; N = s ( 1)n Hn(x) :
n
n n
p
n
2 (n!) 1 p
46
2.22 Laguerre ! Hermite
The Hermite polynomials dened by (1.13.1) can be obtained from the Laguerre polynomials given
by (1.11.1) by taking the limit ! 1 in the following way :
n
lim 2
!1 2
n
L(n) (2) 21 x + = ( n1)! Hn(x):
2.23 Charlier ! Hermite
If we set x ! (2a)1=2x + a in the denition (1.12.1) of the Charlier polynomials and let a ! 1
we nd the Hermite polynomials dened by (1.13.1). In fact we have
lim (2a) n2 Cn (2a) 12 x + a; a = ( 1)n Hn(x):
a!1
47
SCHEME
OF
BASIC HYPERGEOMETRIC
ORTHOGONAL POLYNOMIALS
Askey-Wilson
(4)
Continuous
dual q-Hahn
(3)
(2)
(1)
(0)
Al-Salam
Chihara
Continuous
big q-Hermite
q-Meixner
Pollaczek
Big
q-Jacobi
Continuous
q-Hahn
Continuous
q-Jacobi
Continuous
q-Laguerre
Big
q-Laguerre
Little
q-Laguerre
Stieltjes
Wigert
Continuous
q-Hermite
48
Little
q-Jacobi
q-Laguerre
SCHEME
OF
BASIC HYPERGEOMETRIC
ORTHOGONAL POLYNOMIALS
q-Racah
Big
q-Jacobi
(4)
q-Hahn
q-Meixner
Alternative
q-Charlier
Quantum
q-Krawtchouk
Dual q-Hahn
q-Krawtchouk
Ane
q-Krawtchouk
Al-Salam
Carlitz I
q-Charlier
Discrete
q-Hermite I
Discrete
q-Hermite II
49
(3)
Dual
q-Krawtchouk
Al-Salam
Carlitz II
(2)
(1)
(0)
Chapter 3
Basic hypergeometric orthogonal
polynomials
3.1 Askey-Wilson
Denition.
anpn (x; a; b; c; djq) = q n ; abcdqn 1; aei ; ae i q; q ; x = cos :
(3.1.1)
4 3
ab; ac; ad
(ab; ac; ad; q)n
The Askey-Wilson polynomials are q-analogues of the Wilson polynomials given by (1.1.1).
Orthogonality. When a; b; c; d are real, or occur in complex conjugate pairs if complex, and
max(jaj; jbj; jcj; jdj) < 1, then we have the following orthogonality relation
1 Z pw(x) p (x; a; b; c; djq)p (x; a; b; c; djq)dx = h ;
m
n
n mn
2
2
1
x
1
1
where
(
(3.1.2)
2i
2
21 )h(x; q 21 )
(
e
;
q
)
h
(
x;
1)
h
(
x;
1)
h
(
x;
q
1
w(x) := w(x; a; b; c; djq) = aei ; bei ; cei ; dei ; q) = h(x; a)h(x; b)h(x; c)h(x; d) ;
1
with
1 Y
h(x; ) :=
1 2xqk + 2q2k = ei ; e i ; q 1 ; x = cos k=0
and
(abcdqn 1; q)n(abcdq2n; q)1
hn = (qn+1 ; abq
n; acqn; adqn; bcqn; bdqn; cdqn; q)1 :
If a > 1 and b; c; d are real or one is real and the other two are complex conjugates,
max(jbj; jcj; jdj) < 1 and the pairwise products of a; b; c and d have absolute value less than one,
then we have another orthogonality relation given by :
1 Z pw(x) p (x; a; b; c; djq)p (x; a; b; c; djq)dx +
n
2
1 x2 m
1
1
X
+
1
k
<aqk a
wk pm (xk ; a; b; c; djq)pn(xk ; a; b; c; djq) = hn mn ;
50
(3.1.3)
where w(x) and hn are as before,
and
aqk + aqk
xk =
2
1
2
2 2k
2
ad; q)k q k :
wk = (q; ab; ac; ad;(aa 1b;; qa)11c; a 1d; q) (1 (1a2)(aq;qab )(1aq;;acab;1ac;
q; ad 1q; q)k abcd
1
Recurrence relation.
2xp~n(x) = Anp~n+1 (x) + a + a
where
1
(An + Cn) p~n (x) + Cn p~n 1(x);
n n(x; a; b; c; djq)
p~n(x) := a (pab;
ac; ad; q)
n
and
8
>
>
>
>
<
n
acqn)(1 adqn)(1 abcdqn 1)
An = (1 abqa(1)(1 abcdq
2n 1)(1
abcdq2n)
>
>
>
>
:
n
bcqn 1)(1 bdqn 1)(1 cdqn 1) :
Cn = a(1 q (1)(1 abcdq
2n 2)(1
abcdq2n 1)
q-Dierence equation.
h
i
(1 q)2 Dq w~(x; aq 21 ; bq 21 ; cq 21 ; dq 21 jq)Dq y(x) +
+ nw~(x; a; b; c; djq)y(x) = 0; y(x) = pn (x; a; b; c; djq);
where
If we dene
Dq f (x) := qf (xx) with q f (ei ) = f (q 12 ei ) f (q 21 ei ); x = cos :
q
n
n 1; az; az 1 q; q
Pn(z ) := (ab; ac;anad; q)n 4 3 q ; abcdq
ab; ac; ad
then the q-dierence equation can also be written in the form
q n (1 qn)(1 abcdqn 1)Pn (z )
= A(z )Pn (qz ) A(z ) + A(z 1 ) Pn(z ) + A(z 1 )Pn(q 1 z );
where
bz )(1 cz )(1 dz ) :
A(z ) = (1 az )(1
(1 z 2)(1 qz 2 )
Generating functions.
aei ; bei q; e i t ce i ; de
2 1
2 1
cd
ab 1
(3.1.5)
w~(x; a; b; c; djq) := w(xp; a; b; c;2djq) ;
1 x
n
+1
n
n = 4q (1 q )(1 abcdqn 1)
and
2
(3.1.4)
aei ; cei q; e i t be i ; de
2 1
ac bd
i i q; ei t =
q; ei t
51
=
(3.1.6)
1
X
pn (x; a; b; c; djq) tn ; x = cos : (3.1.7)
n=0 (ab; cd; q; q)n
1
X
pn (x; a; b; c; djq) tn ; x = cos : (3.1.8)
n=0 (ac; bd; q; q)n
1 p (x; a; b; c; djq)
i ; dei i ; ce i X
ae
be
n
i
i
n
2 1
ad q; e t 21
bc q; e t = n=0 (ad; bc; q; q)n t ; x = cos : (3.1.9)
Remark. The q-Racah polynomials dened by (3.2.1) and the Askey-Wilson polynomials
given by (3.1.1) are related in the following way. If we substitute a2 = q, b2 = 2 1 1 q,
c2 = 2 1 q, d2 = 1 q and e2i = q2x+1 in the denition (3.1.1) of the Askey-Wilson
polynomials we nd :
12 n
12 21 12
21 12 q 21 ; 21 21 q 12 ; 12 21 q 21 jq)
(
q
)
p
(
(
x
);
q
;
n
Rn ((x); ; ; ; jq) =
;
(q; q; q; q)n
where
(x) = 21 21 21 qx+ 12 + 21 12 12 q x 12 :
References. [10], [25], [31], [45], [47], [48], [73], [112], [114], [127], [131], [133], [134], [136],
[137], [140], [162], [163], [166], [176], [179], [180], [197], [198], [200], [202], [228].
3.2
q
-Racah
Denition.
n
n+1 ; q x; qx+1 q
;
q
~
q; q ; n = 0; 1; 2; : : :; N;
Rn((x); ; ; ; jq) = 4 3
q; q; q
where
(x) := q x + qx+1
and
q = q N or q = q N or q = q N ; with N a nonnegative integer.
Since
kY1
(q x ; qx+1 ; q)k =
1 (x)qj + q2j +1 ;
(3.2.1)
j =0
it is clear that Rn ((x); ; ; ; jq) is a polynomial of degree n in (x).
Orthogonality.
N
X
where
(q; q; q; q; q)x (1 q2x+1 ) R ((x))R ((x)) = h ;
m
n
n mn
1
1
x
x=0 (q; q; q; q; q)x (q) (1 q)
(3.2.2)
Rn((x)) := Rn((x); ; ; ; jq)
and
2
; 1 1 ; 1 ; 1 ; q)1 (1 q)(q)n (q; q; 1q; 1 q; q)n :
hn = ( (q
1 q; 1 q; q; 1 1 q 1 ; q )
1 (1 q2n+1 ) (q; q; q; q; q)n
This implies
8
(q2 ; 1 ; q)N (1 q N )(q)n (q; q; 1 q N ; 1 q N ; q)n
>
>
if q = q N
>
>
1 q; q ; q )
2n N )
N ; q N ; q; q; q)n
>
(
(1
q
(
q
N
>
>
>
>
>
<
1 ; q2; q)N (1 q)( 1 q N )n (q; q; qN +2 ; 1 q; q)n if q = q
hn = > ((
1 q; q ; q )
(1 q2n+1 )
(q; q; q N ; q; q)n
N
>
>
>
>
>
>
>
>
>
:
(q2 ; 1; q)N (1 q)(q N )n (q; q; 1q; qN +2 ; q)n
(q; 1 q; q)N (1 q2n+1 ) (q; q; q; q N ; q)n
52
N
if q = q N :
Recurrence relation.
1 q x 1 qx+1 Rn((x))
= An Rn+1((x)) (An + Cn) Rn((x)) + CnRn 1((x));
where
8
>
>
>
>
<
>
>
>
>
:
(3.2.3)
n+1 )(1 qn+1 )(1 qn+1 )(1 qn+1 )
An = (1 q (1
q2n+1 )(1 q2n+2 )
n
n )( qn)( qn )
Cn = q(1 q(1)(1 qq2n)(1
:
q2n+1 )
q-Dierence equation.
[w(x 1)B (x 1)y(x 1)] +
q n (1 qn )(1 qn+1 )w(x)y(x) = 0; y(x) = Rn((x); ; ; ; jq); (3.2.4)
where
f (x) := f (x + 1) f (x);
q; q; q; q)x (1 q2x+1 )
w(x) = (q;(q;1q;
1 q; q; q)x (q)x (1 q)
and B (x) as below. This q-dierence equation can also be written in the form
q n(1 qn )(1 qn+1 )y(x) = B (x)y(x + 1) [B (x) + D(x)] y(x) + D(x)y(x 1); (3.2.5)
where
y(x) = Rn((x); ; ; ; jq)
and
8
(1 qx+1 )(1 qx+1 )(1 qx+1 )(1 qx+1 )
>
>
>
B
(
x
)
=
>
<
(1 q2x+1 )(1 q2x+2 )
>
>
>
>
:
x
x)( qx )( qx )
D(x) = q(1 q(1)(1 qq2x )(1
:
q2x+1 )
Generating functions.
x+1
x+1
1
x 1 x
~ q ; q q; q xt 21 q ; q q; qx+1 t
2 1
q
q
N (q; q; q)
X
n R ((x); ; ; ; jq)tn:
'
n
(
q;
q
;
q
)
n
n=0
~
2 1
(3.2.6)
qx+1 ; qx+1 q; q xt 1 1 q x; 1 q x q; qx+1 t
2 1
1 q
q
N (q; q; q)
X
n R ((x); ; ; ; jq)tn:
' (
n
1 q; q ; q )
n
n=0
(3.2.7)
x+1
q ; qx+1 q; q xt 1 1 q x ; 1 q x q; qx+1 t
~
2 1
2 1
1 q
q
N (q; q; q)
X
n R ((x); ; ; ; jq)tn:
' (
n
1 q; q ; q )
n
n=0
53
(3.2.8)
Remark. The Askey-Wilson polynomials dened by (3.1.1) and the q-Racah polynomials given
by (3.2.1) are related in the following way. If we substitute = abq 1 , = cdq 1, = adq 1,
= ad 1 and qx = a 1 e i in the denition (3.2.1) of the q-Racah polynomials we nd :
(x) = 2a cos and
n n (x; a; b; c; djq)
Rn 2a cos ; abq 1; cdq 1; adq 1; ad 1jq = a (pab;
ac; ad; q) :
n
References. [10], [22], [25], [43], [45], [111], [114], [127], [160], [180], [183], [197].
3.3 Continuous dual q-Hahn
Denition.
an pn(x; a; b; cjq) = q n; aei ; ae i q; q ; x = cos :
(3.3.1)
3 2
ab; ac (ab; ac; q)n
Orthogonality. When a; b; c are real, or one is real and the other two are complex conjugates,
and max(jaj; jbj; jcj) < 1, then we have the following orthogonality relation
1 Z pw(x) p (x; a; b; cjq)p (x; a; b; cjq)dx = h ;
n
n mn
2
1 x2 m
1
1
where
with
(3.3.2)
(
2i
2
x; 1)h(x; q 21 )h(x; q 21 ) ;
w(x) := w(x; a; b; cjq) = aei ;(ebei;; qce)1i ; q) = h(x; 1)hh((x;
a)h(x; b)h(x; c)
1
h(x; ) :=
and
1 Y
k=0
1 2xqk + 2q2k = ei ; e i ; q 1 ; x = cos 1
hn = (qn+1 ; abqn; acq
n; bcqn; q) :
1
If a > 1 and b and c are real or complex conjugates, max(jbj; jcj) < 1 and the pairwise products
of a; b and c have absolute value less than one, then we have another orthogonality relation given
by :
1 Z pw(x) p (x; a; b; cjq)p (x; a; b; cjq)dx +
n
2
1 x2 m
1
1
X
+
1
k
<aqk a
where w(x) and hn are as before,
and
wk pm (xk ; a; b; cjq)pn(xk ; a; b; cjq) = hn mn ;
k
k
xk = aq + 2aq
1
k
2
2 2k
2
k)
(
a
;
q
)
(1
a
q
)(
a
;
ab;
ac
;
q
)
1
1
k
k
(
2
wk = (q; ab; ac; a 1b; a 1c; q) (1 a2 )(q; ab 1q; ac 1q; q) ( 1) q
a2bc :
1
k
54
(3.3.3)
Recurrence relation.
2xp~n(x) = Anp~n+1 (x) + a + a
where
1
(An + Cn) p~n (x) + Cn p~n 1(x);
(3.3.4)
n n(x; a; b; cjq)
p~n(x) := a p(ab;
ac; q)n
and
q-Dierence equation.
8
<
An = a 1 (1 abqn )(1 acqn)
:
Cn = a(1 qn )(1 bcqn 1):
h
i
(1 q)2 Dq w~ (x; aq 21 ; bq 21 ; cq 21 jq)Dq y(x) +
+ 4q n+1 (1 qn )w~ (x; a; b; cjq)y(x) = 0; y(x) = pn(x; a; b; cjq); (3.3.5)
where
w~(x; a; b; cjq) := w(px; a; b; c2jq)
1 x
and
If we dene
Dq f (x) := qf (xx) with q f (ei ) = f (q 12 ei ) f (q 21 ei ); x = cos :
q
n
(
ab;
ac
;
q
)
az; az
n
Pn(z ) := an 32 q ;ab;
ac
then the q-dierence equation can also be written in the form
q n (1 qn)Pn (z ) = A(z )Pn (qz )
where
Generating functions.
1
q; q
A(z ) + A(z 1 ) Pn(z ) + A(z 1 )Pn(q 1 z );
(3.3.6)
bz )(1 cz )
A(z ) = (1 (1az )(1
z 2)(1 qz 2 ) :
1 p (x; a; b; cjq)
(ct; q)1 aei ; bei q; e i t = X
n
tn; x = cos :
ab (ei t; q)1 2 1
(
ab;
q
;
q
)
n
n=0
(3.3.7)
1 p (x; a; b; cjq)
(bt; q)1 aei ; cei q; e i t = X
n
tn ; x = cos :
2
1
ac (ei t; q)1
(
ac;
q
;
q
)
n
n=0
(3.3.8)
1 p (x; a; b; cjq)
(at; q)1 bei ; cei q; e i t = X
n
tn; x = cos :
2
1
bc (ei t; q)1
(
bc;
q
;
q
)
n
n=0
(3.3.9)
References.
3.4 Continuous q-Hahn
Denition.
(aei )npn (x; a; b; c; d; q) = q n ; abcdqn 1; aei(+2); ae i q; q ; x = cos( + ): (3.4.1)
4 3
abe2i ; ac; ad
(abe2i ; ac; ad; q)n
55
Orthogonality. When c = a and d = b then we have, if a and b are real and max(jaj; jbj) < 1
or if b = a and jaj < 1 :
1 Z w(cos( + ))p (cos( + ); a; b; c; d; q)p (cos( + ); a; b; c; d; q)d = h ;
m
n
n mn
4
(3.4.2)
where
2
2i( +)
w(x) := w(x; a; b; c; d; q) = (aei(+2) ;(bee i(+2);;qce)1i ; dei ; q) 1
1
1
2 )h(x; q 2 )
h
(
x;
1)
h
(
x;
1)
h
(
x;
q
= h(x; aei)h(x; bei)h(x; ce i )h(x; de i) ;
with
h(x; ) :=
and
1 Y
k=0
1 2xqk + 2 q2k = ei(+) ; e i(+); q 1 ; x = cos( + )
n 1; q)n(abcdq2n; q)1
hn = (qn+1 ; abqne2(iabcdq
; acqn; adqn; bcqn; bdqn; cdqne
2
Recurrence relation.
i ; q)1 :
2xp~n(x) = An p~n+1(x) + aei + a 1e i (An + Cn) p~n (x) + Cnp~n 1(x);
where
(3.4.3)
i n pn(x; a; b; c; d; q)
p~n(x) := (ae(ac;) ad;
abe2i; q)
n
and
8
>
>
>
>
<
2i n
acqn)(1 adqn)(1 abcdqn 1)
An = (1 abe aeiq (1)(1 abcdq
2n 1)(1
abcdq2n)
>
>
>
>
:
i
n
bcqn 1)(1 bdqn 1)(1 cde
Cn = ae (1 q )(1
(1 abcdq2n 2)(1 abcdq2n 1)
q-Dierence equation.
2
iqn
1
):
i
h
(1 q)2 Dq w~(x; aq 21 ; bq 21 ; cq 12 ; dq 21 ; q)Dq y(x) +
+ n w~(x; a; b; c; d; q)y(x) = 0; y(x) = pn (x; a; b; c; d; q);
where
and
(3.4.4)
w~(x; a; b; c; d; q) := w(xp; a; b; c;2d; q) ;
1 x
n
+1
n
n = 4q (1 q )(1 abcdqn 1)
Dq f (x) := qf (xx) with q f (ei(+) ) = f (q 12 ei(+) ) f (q 12 ei(+) ); x = cos( + ):
q
Generating functions.
2 1
aei(+2) ; bei(+2) q; e i(+)t ce i(+2) ; de i(+2) q; ei(+)t
2 1
cde 2i
abe2i
1 p (x; a; b; c; d; q)tn
X
n
; x = cos( + ):
(3.4.5)
= (abe
2i ; cde 2i ; q ; q )
n
n=0
56
i(+2)
ae
; cei q; e i(+) t be i ; de i(+2) q; ei(+)t
2 1
2 1
ac
bd
1
X
x; a; b; c; d; q) tn; x = cos( + ):
= pn((ac;
bd; q; q)n
n=0
2
1
(3.4.6)
aei(+2) ; dei q; e i(+)t be i ; ce i(+2) q; ei(+)t
2 1
bc
ad
1
X
x; a; b; c; d; q) tn; x = cos( + ):
= pn((ad;
bc; q; q)n
n=0
(3.4.7)
References. [25], [45], [114].
3.5 Big q-Jacobi
Denition.
Pn(x; a; b; c; q) = 3 2
Orthogonality.
Zaq
cq
q n; abqn+1; x q; q :
aq; cq (3.5.1)
(a 1 x; c 1x; q)1 P (x; a; b; c; q)P (x; a; b; c; q)d x
n
q
(x; bc 1x; q)1 m
a 1c; ac 1q; abq2; q)1 = aq(1 q) (q;(aq;
bq; cq; abc 1q; q)1
) (q; bq; abc 1q; q)n ( acq2 )n q(n2 ) :
(1 (1 abqabq
mn
2n+1 ) (abq; aq; cq ; q )
n
(3.5.2)
Recurrence relation.
(x 1)Pn(x; a; b; c; q)
= An Pn+1(x; a; b; c; q) (An + Cn) Pn(x; a; b; c; q) + CnPn 1(x; a; b; c; q);
where
8
>
>
>
>
<
n+1
n+1
abqn+1)
An = (1 (1aq abq)(12n+1cq)(1 )(1
abq2n+2)
>
>
>
>
:
n
bqn)(1 abc 1 qn) :
Cn = acqn+1 (1 (1q )(1
abq2n)(1 abq2n+1)
q-Dierence equation.
q n (1 qn )(1 abqn+1)x2 y(x) = B (x)y(qx) [B (x) + D(x)] y(x) + D(x)y(q 1 x);
where
y(x) = Pn (x; a; b; c; q)
and
8
< B (x) = aq(x 1)(bx c)
:
D(x) = (x aq)(x cq):
57
(3.5.3)
(3.5.4)
Generating functions.
2
2
1
1 (cq; q)
aqx 1 ; 0 q; xt bc 1x q; cqt = X
n P (x; a; b; c; q)tn:
1
1
aq bq (
bq;
q
;
q
)n n
n=0
(3.5.5)
1
(aq; q)n P (x; a; b; c; q)tn:
cqx 1; 0 q; xt bc 1 x q; aqt = X
n
1
1
1
1
abc q
cq
n=0 (abc q; q; q)n
(3.5.6)
1
Remarks. The big q-Jacobi polynomials with c = 0 and the little q-Jacobi polynomials dened
by (3.12.1) are related in the following way :
bq; q)n ( 1)n anqn+(n2 ) p x ; b; a q :
Pn (x; a; b; 0; q) = ((aq
n aq
; q)n
Sometimes the big q-Jacobi polynomials are dened in terms of four parameters instead of
three. In fact the polynomials given by the denition
n
abqn+1; ac 1qx q; q
Pn(x; a; b; c; d; q) = 3 2 q ;aq;
ac 1dq are orthogonal on the interval [ d; c] with respect to the weight function
(c 1 qx; d 1qx; q)1 d x:
(ac 1 qx; bd 1qx; q)1 q
These polynomials are not really dierent from those dened by (3.5.1) since we have
Pn (x; a; b; c; d; q) = Pn(ac 1qx; a; b; ac 1d; q)
and
Pn(x; a; b; c; q) = Pn(x; a; b; aq; cq; q):
References. [8], [10], [25], [114], [121], [127], [140], [142], [160], [163], [176], [180], [181], [212].
Special case
3.5.1 Big q-Legendre
Denition. The big q-Legendre polynomials are big q-Jacobi polynomials with a = b = 1 :
n n+1 Pn (x; c; q) = 32 q ;q;qcq ; x q; q :
(3.5.7)
Orthogonality.
Zq
cq
) (c 1 q; q)n ( cq2 )n q(n2 ) :
Pm (x; c; q)Pn(x; c; q)dq x = q(1 c) (1 (1 q2nq+1
mn
) (cq; q)n
(3.5.8)
Recurrence relation.
(x 1)Pn(x; c; q) = An Pn+1(x; c; q) (An + Cn ) Pn(x; c; q) + Cn Pn 1(x; c; q);
58
(3.5.9)
where
8
>
>
>
>
<
qn+1 )(1 cqn+1)
An = (1
(1 + qn+1 )(1 q2n+1 )
>
>
>
>
:
qn)(1 c 1 qn) :
Cn = cqn+1 (1
(1 + qn)(1 q2n+1)
q-Dierence equation.
q n(1 qn)(1 qn+1 )x2y(x) = B (x)y(qx) [B (x) + D(x)] y(x) + D(x)y(q 1 x);
where
y(x) = Pn(x; c; q)
and
8
< B (x) = q(x 1)(x c)
:
Generating functions.
2 1
(3.5.11)
1 P (x; c; q)
cqx 1 ; 0 q; xt c 1x q; qt = X
n
tn:
1
1
1 q; q)
cq c 1q (
c
n
n=0
(3.5.12)
References. [160].
q
D(x) = (x q)(x cq):
1 (cq; q)
qx 1; 0 q; xt c 1x q; cqt = X
n P (x; c; q)tn:
n
1 1
q (
q;
q
;
q
)
q n
n=0
2 1
3.6
(3.5.10)
-Hahn
Denition.
n
qn+1 ; q
Qn (q x ; ; ; N jq) = 3 ~2 q ; q;
q N
x q; q ; n = 0; 1; 2; : ::; N:
(3.6.1)
Orthogonality.
N
X
(q; q N ; q)x (q) x Q (q x ; ; ; N jq)Q (q x ; ; ; N jq)
m
n
1
N
x=0 (q; q ; q)x
2
; q)N (q; q; qN +2 ; q)n (1 q)( q)n q(n2 ) Nn :
= (q(q
mn
; q)N (q)N (q; q; q N ; q)n (1 q2n+1 )
(3.6.2)
Recurrence relation.
1 q x Qn(q x ) = An Qn+1(q x ) (An + Cn) Qn(q x ) + CnQn 1(q x );
where
and
Qn(q x ) := Qn (q x ; ; ; N jq)
8
>
>
>
>
<
n N )(1 qn+1 )(1 qn+1 )
An = (1 (1q q
2n+1 )(1
q2n+2 )
>
>
>
>
:
n
n)(1 qn )(q N qn+1 )
:
Cn = q (1(1 q q
2n )(1
q2n+1 )
59
(3.6.3)
q-Dierence equation.
q n(1 qn )(1 qn+1 )y(x) = B (x)y(x + 1) [B (x) + D(x)] y(x) + D(x)y(x 1); (3.6.4)
where
y(x) = Qn(q x ; ; ; N jq)
and
8
x N )(1 qx+1 )
< B (x) = (1 q
:
Generating functions.
2
1
D(x) = q(1 qx )( qx
N
1
):
N (q N ; q)
qx N ; 0 q; q xt q x q; qt ' X
n Q (q x ; ; ; N jq)tn:
1
1
n
q q (
q;
q
;
q
)
n
n=0
(3.6.5)
N +1 x
x N
~ q N; 0 q; q xt 1 1 q N +2 q; qt
2 1
q
q
N
X
; q)n
x
n
' (q(q
N +2 ; q; q)n Qn(q ; ; ; N jq)t :
n=0
(3.6.6)
Remarks. The q-Hahn polynomials dened by (3.6.1) and the dual q-Hahn polynomials given
by (3.7.1) are related in the following way :
Qn(q x ; ; ; N jq) = Rx((n); ; ; N jq);
with
(n) = q n + qn+1
or
Rn((x); ; ; N jq) = Qx(q n ; ; ; N jq);
where
(x) = q x + qx+1 :
For x = 0; 1; 2; : ::; N the generating function (3.6.5) can also be written as :
x N
N (q N ; q)
q ; 0 q; q xt q x q; qt = X
n Q (q x ; ; ; N jq)tn:
n
1 1
2 1
q (
q;
q
;
q
)
q n
n=0
References. [10], [25], [43], [45], [88], [111], [114], [121], [142], [145], [158], [160], [180], [197],
[215], [217], [218].
3.7 Dual q-Hahn
Denition.
n
q x ; qx+1 q; q ; n = 0; 1; 2; : ::; N;
Rn((x); ; ; N jq) = 3 ~2 q ;q;
q N where
(x) := q x + qx+1 :
60
(3.7.1)
Orthogonality.
N
X
(q; q; q N ; q)x (1 q2x+1 ) qNx (x2) R ((x); ; ; N jq)R ((x); ; ; N jq)
m
n
N +2 ; q; q)x (1 q)( q)x
x=0 (q; q
2
; q)N (q) N (q; 1q N ; q)n (q)n :
= (q
(3.7.2)
mn
(q; q)N
(q; q N ; q)n
Recurrence relation.
1 q x 1 qx+1 Rn((x))
= An Rn+1((x)) (An + Cn) Rn((x)) + CnRn 1((x));
where
(3.7.3)
Rn((x)) := Rn((x); ; ; N jq)
and
8
<
An = 1 q n
:
Cn = q (1 qn) qn
N
1 qn+1
N
1
:
q-Dierence equation.
q n(1 qn )y(x) = B (x)y(x + 1) [B (x) + D(x)] y(x) + D(x)y(x 1);
where
y(x) = Rn((x); ; ; N jq)
and
8
(1 qx+1 )(1 qx+1 )(1 qx N )
>
>
B
(
x
)
=
>
>
<
(1 q2x+1 )(1 q2x+2 )
>
>
>
>
:
(3.7.4)
x N
x )(1 qx )(1 qx+N +1 )
:
D(x) = q (1(1 qq
2x )(1
q2x+1 )
Generating functions.
N (q; q)
(qt; q)1 ~ qx N ; qx+1 q; q xt ' X
n R ((x); ; ; N jq)tn:
1
n
2
q N (qx+1 t; q)1
(
q
;
q
)
n
n=0
(3.7.5)
N (q N ; q)
(q N t; q)1 q x ; 1q x q; qx+1 t ' X
n R ((x); ; ; N jq)tn :
2
1
n
q
(q x t; q)1
(
q
;
q
)
n
n=0
(3.7.6)
N (q N ; q; q)
(qt; q)1 qx N ; qx+1 q; q x t ' X
n R ((x); ; ; N jq)tn: (3.7.7)
n
1 q N ; q; q)
(qx+1 t; q)1 2 1 1q N (
n
n=0
Remarks. The dual q-Hahn polynomials dened by (3.7.1) and the q-Hahn polynomials given
by (3.6.1) are related in the following way :
Qn(q x ; ; ; N jq) = Rx((n); ; ; N jq);
with
(n) = q n + qn+1
or
Rn((x); ; ; N jq) = Qx(q n ; ; ; N jq);
61
where
(x) = q x + qx+1 :
For x = 0; 1; 2; : ::; N the generating function (3.7.6) can also be written as :
(q
N t; q)N x 2 1
q x ; 1q
q
x q; qx+1 t =
N
X
(q N ; q)n R ((x); ; ; N jq)tn:
n
n=0 (q; q)n
For x = 0; 1; 2; : ::; N the generating function (3.7.7) can also be written as :
(qt; q)x 2 1
N (q N ; q; q)
qx N ; qx+1 q; q xt = X
n R ((x); ; ; N jq)tn:
n
1 q N ; q; q)
1q N (
n
n=0
References. [25], [43], [45], [114], [145], [180], [217].
3.8 Al-Salam-Chihara
Denition.
n
i ; ae i q
;
ae
(
ab
;
q
)
n
q; q
(3.8.1)
Qn(x; a; bjq) = an 3 2
ab; 0
n i = (be i ; q)nein 2 1 bq 1q1; aenei q; b 1qe i ; x = cos :
Orthogonality. When a and b are real or complex conjugates and max(jaj; jbj) < 1, then we
have the following orthogonality relation
1 Z pw(x) Q (x; a; bjq)Q (x; a; bjq)dx =
mn
m
n
2
2
(qn+1; abqn; q)1 ;
1
x
1
1
where
(
(3.8.2)
12
2i
2
21
w(x) := w(x; a; bjq) = ae(ie ; be; iq);1q) = h(x; 1)h(x;h(x;1)ah)(hx;(x;q b))h(x; q ) ;
1
with
h(x; ) :=
1 Y
k=0
1 2xqk + 2q2k = ei ; e i ; q 1 ; x = cos :
If a > 1, jbj < 1 and jabj < 1, then we have another orthogonality relation given by :
1 Z pw(x) Q (x; a; bjq)Q (x; a; bjq)dx +
n
2
1 x2 m
1
1
X
+
1
k
<aqk a
where w(x) is as before,
mn
wk Qm (xk ; a; bjq)Qn(xk ; a; bjq) = (qn+1 ;abq
n; q) ;
1
k
k
xk = aq + 2aq
62
1
(3.8.3)
and
(a 2 ; q)1 (1 a2q2k )(a2 ; ab; q)k q
wk = (q; ab;
a 1b; q)1 (1 a2 )(q; ab 1q; q)k
Recurrence relation.
2xQ~ n(x) = AnQ~ n+1 (x) + a + a
where
k2
1 k:
a3 b
(An + Cn) Q~ n (x) + Cn Q~ n 1(x);
1
(3.8.4)
n
bjq)
Q~ n(x) := a Q(nab(x; ;qa;
)n
and
8
<
An = a 1 (1 abqn)
:
Cn = a(1 qn):
q-Dierence equation.
h
i
(1 q)2 Dq w~(x; aq 21 ; bq 21 jq)Dq y(x) +
+ 4q n+1(1 qn)w~(x; a; bjq)y(x) = 0; y(x) = Qn(x; a; bjq);
where
(3.8.5)
w~ (x; a; bjq) := wp(x; a; bj2q)
1 x
and
If we dene
Dq f (x) := qf (xx) with q f (ei ) = f (q 12 ei ) f (q 21 ei ); x = cos :
q
n az; az 1 Pn(z ) := (aba;nq)n 3 2 q ;ab;
0 q; q
then the q-dierence equation can also be written in the form
q n (1 qn)Pn (z ) = A(z )Pn (qz )
where
Generating functions.
A(z ) + A(z 1 ) Pn(z ) + A(z 1 )Pn(q 1 z );
)(1 bz ) :
A(z ) = (1(1 zaz2)(1
qz 2 )
1 Q (x; a; bjq)
aei ; bei q; e i t = X
1
n
tn; x = cos :
2
1
ab (ei t; q)1
(
ab;
q
;
q
)
n
n=0
(3.8.6)
1 Q (x; a; bjq)
(at; bt; q)1 = X
n
n
(ei t; e i t; q)1 n=0 (q; q)n t ; x = cos :
References. [10], [15], [16], [39], [79], [84].
63
(3.8.7)
(3.8.8)
3.9
q
-Meixner-Pollaczek
Denition.
n
i(+2); ae i q
;
ae
;
q
)
n
q; q
(3.9.1)
Pn(x; ajq) = a
a2 ; 0 (q; q)n 3 2
n i i
= (ae(q; q;) q)n ein(+) 2 1 aq 1q1; aenei q; qa 1e i(+2) ; x = cos( + ):
n
n e in (a
2
Orthogonality.
1 Z w(cos( + ); ajq)P (cos( + ); ajq)P (cos( + ); ajq)d =
mn
m
n
2
(q; q)n (q; a2qn ; q)1 ; (3.9.2)
where
0<a<1
and
(
2
1
1
2i( +)
2 )h(x; q 2 )
h
(
x;
1)
h
(
x;
1)
h
(
x;
q
(
e
;
q
)
1
w(x; ajq) = aei(+2) ; aei ; q) =
;
h(x; aei)h(x; ae i )
1
with
h(x; ) :=
1
Y
1 2xqk + 2 q2k = ei(+) ; e i(+) ; q ; x = cos( + ):
1
k=0
Recurrence relation.
2xPn(x; ajq) = (1 qn+1)Pn+1 (x; ajq) +
+ 2aqn cos Pn(x; ajq) + (1 a2qn 1)Pn 1(x; ajq):
(3.9.3)
q-Dierence equation.
h
i
(1 q)2 Dq w~(x; aq 21 jq)Dq y(x) + 4q n+1(1 qn )w~ (x; ajq)y(x) = 0; y(x) = Pn(x; ajq); (3.9.4)
where
and
w~(x; ajq) := wp(x; ajq2)
1 x
Dq f (x) := qf (xx) with q f (ei(+) ) = f (q 12 ei(+) ) f (q 12 ei(+) ); x = cos( + ):
q
Generating functions.
1
ae t q 2 = X
n
e
t q 1 n=0 Pn(x; ajq)t ; x = cos( + ):
( i ; )1
( i(+) ; )
1
(ei(+) t; q)1 1
2
aei(+2); aei q; e
a2
i(+) t
References. [10], [16], [39], [45], [72], [126].
64
=
1
X
(3.9.5)
Pn (x; ajq) tn; x = cos( + ): (3.9.6)
2
n=0 (a ; q)n
3.10 Continuous q-Jacobi
Denitions. If we take a = q 21 14 , b = q 12 43 , c = q 21 41 and d = q 12 34 in the denition
+
+
+
(3.1.1) of the Askey-Wilson polynomials we nd after renormalizing
+
!
n ; qn+++1 ; q 21 + 41 ei ; q 21 + 14 e i +1 ; q)n
q
(
q
Pn(;) (xjq) = (q; q) 43
q+1 ; q 21 (++1) ; q 21 (++2) q; q ; x = cos : (3.10.1)
n
In [196] M. Rahman takes a = q 12 , b = q+ 12 , c = q+ 21 and d = q 21 to obtain after renormalizing
Pn;)(x; q) =
(
!
(q+1; q+1 ; q)n q n ; qn+++1 ; q 21 ei ; q 21 e i q; q ; x = cos : (3.10.2)
(q; q; q)n 4 3
q+1 ; q+1 ; q
These two q-analogues of the Jacobi polynomials are not really dierent, since they are connected
by the quadratic transformation :
q)n qn P (;)(x; q):
Pn(;)(xjq2) = ( q(+q;+1
n
; q)n
Orthogonality. For 12 and 21 the orthogonality relations are respectively
1 Z pw(xjq) P (;)(xjq)P (;)(xjq)dx
n
2
1 x2 m
1
1
=
(q 21 (++2) ; q 21 (++3) ; q)1
(q; q+1; q+1 ; q 21 (++1) ; q 21 (++2) ; q)1
++1 )(q+1 ; q+1 ; q 21 (++3) ; q)n
(1 2qn+++1
q(+ 21 )n mn ;
(1 q
)(q; q++1 ; q 21 (++1) ; q)n
where
(3.10.3)
(
(
2
(e2i ; q)1
q 12 + 14 ei ; q 21 + 34 ei ; q 21 + 41 ei ; q 21 + 34 ei ; q)1 2
i ; ei ; q 21 )1
(
e
=
1 + 1 i
1 + 1 i 1
q 2 4 e ; q 2 4 e ; q 2 )1 h(x; 1)h(x; 1)h(x; q 21 )h(x; q 12 )
=
;
12 + 14
h(x; q
)h(x; q 12 + 43 )h(x; q 12 + 41 )h(x; q 21 + 34 )
w(xjq) := w(x; q; q jq) =
with
h(x; ) :=
and
1 Y
k=0
1 2xqk + 2q2k = ei ; e i ; q 1 ; x = cos 1 Z pw(x; q) P (;)(x; q)P (;)(x; q)dx
n
2
1 x2 m
1
(q++2 ; q)1
= (q; q; q+1; q+1
; q+1 ; q+1 ; q++1 ; q)1 ++1 +1 ; q+1; q+1 ; q+1 ; q++1 ; q)n
qnmn ; (3.10.4)
(1 q(1 q2)(n+q++1
)(q++1 ; q; q; q; q; q)n
1
65
where
with
h(x; ) :=
1 Y
k=0
2
(e2i ; q)1
=
1
1
1
1
(q+ 2 ei ; q 2 ei ; q+ 2 ei ; q 2 ei ; q)1 2
i ; ei ; q)1
h(x; 1)h(x; 1) ;
(
e
=
= + 1 i + 1 i
(q 2 e ; q 2 e ; q)1
h(x; q+ 21 )h(x; q+ 12 )
w(x; q) := w(x; q; q ; q)
1 2xqk + 2q2k = ei ; e i ; q 1 ; x = cos :
Recurrence relations.
h
2xP~n(xjq) = AnP~n (xjq) + q 21 41 + q
+
+1
where
1 1
2 4
i
(An + Cn) P~n(xjq) + CnP~n 1(xjq); (3.10.5)
P~n(xjq) := (q(q+1; q;)qn) Pn(;)(xjq)
n
and
8
>
>
>
>
<
n++1)(1 qn+++1 )(1 + qn+ 21 (++1) )(1 + qn+ 21 (++2) )
An = (1 q
q 12 + 14 (1 q2n+++1 )(1 q2n+++2 )
>
>
>
>
:
Cn = q
1 1
2 + 4 (1
qn)(1 qn+ )(1 + qn+ 12 (+) )(1 + qn+ 21 (++1) ) :
(1 q2n++ )(1 q2n+++1 )
h
2xP~n(x; q) = An P~n+1(x; q) + q 21 + q
where
1
2
i
(An + Cn ) P~n(x; q) + CnP~n 1(x; q);
(3.10.6)
(q; q; q)n P (;) (x; q)
P~n(x; q) := (q+1
; q+1 ; q)n n
and
8
>
>
>
>
<
n++1
n+++1
n+1 + qn++1 )
An = (1 q 12 )(1 2nq+++1 )(1 + 2qn++)(1
+2 )
)(1 q
q (1 q
>
>
>
>
:
1
2
qn )(1 qn+ )(1 + qn+ )(1 + qn++ ) :
Cn = q (1 (1
q2n++ )(1 q2n+++1 )
q-Dierence equations.
h
i
(1 q)2 Dq w~ (x; q+ 12 ; q+ 12 jq)Dq y(x) + n w~(x; q; q jq)y(x) = 0; y(x) = Pn(;)(xjq); (3.10.7)
where
x ; q ; q jq ) ;
w~ (x; q; q jq) := w(p
1 x2
n = 4q n+1(1 qn)(1 qn+++1 )
and
Dq f (x) := qf (xx) with q f (ei ) = f (q 12 ei ) f (q 21 ei ); x = cos :
q
h
i
(1 q)2Dq w~(x; q+ 21 ; q+ 21 ; q)Dq y(x) + n w~ (x; q; q ; q)y(x) = 0; y(x) = Pn(;)(x; q); (3.10.8)
66
where
w~ (x; q; q ; q) := w(xp; q ; q 2; q) ;
1 x
n
+1
n
n = 4q (1 q )(1 qn+++1 )
and
Dq f (x) := qf (xx) with q f (ei ) = f (q 12 ei ) f (q 21 ei ); x = cos :
q
Generating functions.
2
2
2
1
1
!
!
!
q 12 + 14 e i ; q 21 + 43 e i q; ei t
q 12 + 41 ei ; q 21 + 34 ei q; e i t 2
1
q+1
q+1
1
1
1
(; )
X
2 (++1)
2 (++2)
= ( q (q+1;; qq+1 ; q) ; q)n Pn( 1 +(1x)jnq) tn; x = cos : (3.10.9)
n
q2 4
n=0
!
q 12 + 41 ei ; q 21 + 14 ei q; e i t q 21 + 34 e i ; q 21 + 43 e
2 1
q 12 (++1) q 12 (++3)
1 ( q 12 (++2) ; q) P (;)(xjq)
X
n n
n
=
1 (++3)
1 + 1 )n t ; x = cos :
(2
2
4
(
q
;
q
)
q
n
n=0
!
q 12 + 41 ei ; q 21 + 34 ei q; e i t q 21 + 34 e i ; q 21 + 41 e
2 1
q 12 (++2) q 12 (++2)
12 (++1)
1
(; )
X
= ( q 1 (++2) ; q)n Pn( 1 +(1x)jnq) tn ; x = cos :
; q)n q 2 4
n=0 ( q 2
1
2
1
!
1
i
q; e t
(3.10.10)
!
i i
q; e t
(3.10.11)
!
q 12 ei ; q+ 21 ei q; e i t q 21 e i ; q+ 12 e i q; ei t
2 1
q+1 q+1
1
X
; q)n Pn(;)(x; q) tn; x = cos :
= (q(+1q;; qq+1
; q)n q 12 n
n=0
2
i !
!
q 12 e i ; q+ 12 e i q; ei t
q 12 ei ; q+ 21 ei q; e i t 2 1
q+1 q+1
1
(; )
X
= ( q(+1q;; qq;q+1)n; q) Pn 1 (nx; q) tn ; x = cos :
n
q2
n=0
!
(3.10.12)
(3.10.13)
!
q 12 ei ; q 21 ei q; e i t q+ 21 e i ; q+ 12 e i q; ei t
2 1
2 1
q
q++1
1
(
;
)
X
q)n Pn (x; q) tn; x = cos :
= ( q(+q;+1
(3.10.14)
; q)n q 12 n
n=0
Remark. The continuous q-Jacobi polynomials given by (3.10.2) and the continuous qultraspherical (or Rogers) polynomials given by (3.10.15) are connected by the quadratic transformations :
1 1
C2n(x; qjq) = (q12 ; q21; q)n q 21 n Pn( 2 ; 2 ) (2x2 1; q)
(q ; q ; q)n
67
and
1 1
C2n+1(x; qjq) = (q12 ; 1;21 q)n+1 q 21 nxPn( 2 ; 2 ) (2x2 1; q):
(q ; q ; q)n+1
References. [45], [112], [114], [136], [179], [180], [196], [198], [199].
Special cases
3.10.1 Continuous q-ultraspherical / Rogers
Denition. If we set a = 12 , b = 12 q 12 , c = 21 and d = 12 q 21 in the denition
(3.1.1) of the Askey-Wilson polynomials and change the normalization we obtain the continuous
q-ultraspherical (or Rogers) polynomials :
!
n 2 qn; 12 ei ; 12 e i 2
Cn(x; jq) = ((q;;qq))n 21 n 43 q ; q
q; q
1
1
2 ; ; q 2
n
n
2i 2
q; q
= ((q;;qq))n n e in 3 2 q ;;2 ; e
0
n
n = ((q;; qq))n ein 2 1 q 1 q;1 n q; 1qe 2i ; x = cos :
(3.10.15)
n
Orthogonality.
1 Z pw(x) C (x; jq)C (x; jq)dx = (; q; q)1 ( 2 ; q)n (1 ) ; j j < 1; (3.10.16)
n
2
( 2 ; q; q)1 (q; q)n (1 qn ) mn
1 x2 m
1
1
where
2
(e2i ; q)1 2
(e2i ; q)1
=
( 21 ei ; 21 q 12 ei ; 12 ei ; 21 q 12 ei ; q)1 (e2i ; q)1 h(x; 1)h(x; 1)h(x; q 21 )h(x; q 12 ) ;
=
h(x; 12 )h(x; 12 q 21 )h(x; 12 )h(x; 21 q 21 )
w(x) := w(x; jq) =
with
h(x; ) :=
1 Y
k=0
1 2xqk + 2q2k = ei ; e i ; q 1 ; x = cos :
Recurrence relation.
2(1 qn )xCn(x; jq) = (1 qn )Cn (x; jq) + (1 qn )Cn (x; jq):
q-Dierence equation.
h
i
(1 q) Dq w~ (x; q 12 jq)Dq y(x) + n w~(x; jq)y(x) = 0; y(x) = Cn(x; jq);
+1
2
+1
1
1
2
where
and
w~(x; jq) := wp(x; jq2) ;
1 x
n = 4q n+1(1 qn )(1 2 qn)
Dq f (x) := qf (xx) with q f (ei ) = f (q 12 ei ) f (q 21 ei ); x = cos :
q
68
(3.10.17)
(3.10.18)
Generating functions.
1
(ei t; e i t; q)1 = X
n
(ei t; e i t; q)1 n=0 Cn(x; jq)t ; x = cos :
(3.10.19)
1 C (x; jq)
1
; e 2i q; ei t = X
n
tn ; x = cos :
2
1
2 ; q)
2 (e i t; q)1
(
n
n=0
(e
i t; q)1 2 1
1 ( 1)n n q(n2 )
; e2i q; e i t = X
Cn(x; jq)tn; x = cos :
2 ; q)
2 (
n
n=0
2
1
1
!
!
!
(3.10.22)
!
12 ei ; 12 ei q; e i t 12 q 21 e i ; 12 q 21 e i q; ei t
2 1
q
1
1
1
X
2 ; q 2 ; q)n
= (q
Cn(x; jq)tn; x = cos :
2;
(
q
;
q
)
n
n=0
(3.10.21)
!
21 e i ; 12 q 21 e i q; ei t
12 ei ; 12 q 21 ei q; e i t 2 1
q 12
q 12
1
1
X
2
= ( ;2 q12 ; q)n Cn(x; jq)tn; x = cos :
n=0 ( ; q ; q)n
2
(3.10.20)
(3.10.23)
!
12 i
12 21 i 21 q 12 e i ; 21 e i e
;
q
e
i
q; e t 21
q; ei t
2 1
q 12
q 12
1 ( ; q 12 ; q)
X
n C (x; jq)tn ; x = cos :
=
(3.10.24)
n
12
2;
(
q
;
q
)
n
n=0
Remarks. The continuous q-ultraspherical (or Rogers) polynomials can also be written as :
Cn (x; jq) =
n
X
( ; q)k ( ; q)n k ei(n 2k); x = cos :
k=0 (q; q)k (q; q)n k
They can be obtained from the continuous q-Jacobi polynomials dened by (3.10.1) in the
following way. Set = in the denition (3.10.1) and change q+ 12 by and we nd the
continuous q-ultraspherical (or Rogers) polynomials with a dierent normalization. We have
+ 12 ! (q 12 ; q)n 1
2n
Pn(;)(xjq) q !
( 2 ; q) Cn (x; jq):
n
If we set = q+ 21 in the denition (3.10.15) of the q-ultraspherical (or Rogers) polynomials
we nd the continuous q-Jacobi polynomials given by (3.10.1) with = . In fact we have
2+1
Cn x; q+ 21 q = +1(q ;(q21)n+ 41 )n Pn(;)(xjq):
(q ; q)nq
If we change q to q 1 we nd
Cn(x; jq 1) = (q)n Cn (x; 1jq):
69
The special case = q of the continuous q-ultraspherical (or Rogers) polynomials equals the
Chebyshev polynomials of the second kind dened by (1.8.21). In fact we have
n + 1) = U (x); x = cos :
Cn (x; qjq) = sin(sin
n
The limit case ! 1 leads to the Chebyshev polynomials of the rst kind given by (1.8.20) in the
following way :
1 qn C (x; jq) = cos n = T (x); x = cos ; n = 1; 2; 3; : :::
lim
n
!1 2(1 ) n
The continuous q-Jacobi polynomials given by (3.10.2) and the continuous q-ultraspherical (or
Rogers) polynomials given by (3.10.15) are connected by the quadratic transformations :
1 1
C2n(x; qjq) = (q12 ; q21; q)n q 21 n Pn( 2 ; 2 ) (2x2 1; q)
(q ; q ; q)n
and
1 1
C2n+1(x; qjq) = (q12 ; 1;21 q)n+1 q 21 nxPn( 2 ; 2 ) (2x2 1; q):
(q ; q ; q)n+1
Finally we remark that the continuous q-ultraspherical (or Rogers) polynomials are related to
the continuous q-Legendre polynomials dened by (3.10.25) in the following way :
Cn(x; qjq2) = q 12 n Pn(x; q):
References. [10], [11], [12], [25], [31], [32], [37], [38], [39], [41], [45], [60], [62], [63], [107], [108],
[109], [110], [112], [113], [114], [127], [137], [159], [179], [180], [181], [202], [204], [207], [208], [209].
3.10.2 Continuous q-Legendre
Denition. The continuous q-Legendre polynomials are continuous q-Jacobi polynomials with
= = 0. If we set = = 0 in the denition (3.10.2) of the continuous q-Jacobi polynomials
we obtain
!
n ; qn+1; q 12 ei ; q 21 e i q
(3.10.25)
Pn(x; q) = 43
q; q ; x = cos :
q; q; q
If we set = = 0 in the denition (3.10.1) we nd
Pn(xjq) = 4 3
q n ; qn+1; q 41 ei ; q 41 e
q; q 21 ; q
i !
q; q ; x = cos ;
but these are not really dierent in view of the quadratic transformation
Pn(xjq2 ) = Pn(x; q):
Orthogonality.
1 Z pw(x) P (x; q)P (x; q)dx = (q; q)2n(q2n+2; q)1 qn ;
n
2
( q; q)41 (q; q)31 mn
1 x2 m
1
1
where
w(x) = (3.10.26)
2
(ei ; ei ; q)1 2
(e2i ; q)1
= h(x; 1)h(x; 1) ;
=
(q 12 ei ; q 21 ei ; q 21 ei ; q 21 ei ; q)1 (q 12 ei ; q 21 ei ; q)1 h(x; q 21 )h(x; q 12 )
70
with
h(x; ) :=
1 Y
k=0
1 2xqk + 2q2k = ei ; e i ; q 1 ; x = cos :
Recurrence relation.
2(1 q2n+1)xPn(x; q) = q 21 (1 q2n+2 )Pn+1(x; q) + q 21 (1 q2n)Pn 1(x; q):
(3.10.27)
q-Dierence equation.
(3.10.28)
(1 q)2Dq [w~(x; q; q)Dq y(x)] + n w~(x; q 21 ; q)y(x) = 0; y(x) = Pn(x; q);
where
n = 4q n+1 (1 qn )(1 qn+1)
and
w~(x; a; q) := wp(x; a; q2) ;
1 x
where
2
1
2i
2
q 12 ) ;
w(x; a; q) = (aei ; aei(;e ae; iq);1 aei ; q) = hh(x;(x;1)ah)h(x;(x; a1))hh((x;x; q a))hh((x;
x; a)
1
with
1 Y
1 2xqk + 2q2k = ei ; e i ; q 1 ; x = cos h(x; ) :=
and
k=0
Dq f (x) := qf (xx) with q f (ei ) = f (q 12 ei ) f (q 21 ei ); x = cos :
q
Generating functions.
2
1
2
1
!
q; ei t
(3.10.29)
!
!
(3.10.30)
!
q 21 ei ; q 23 ei q2; e i t q 12 e i ; q 23 e i q2 ; ei t
2 1
q2 q2
1
n+1
X
= ((q; qq; q;;qq))n Pn (12xn; q) tn ; x = cos :
n q
n=0
2 1
!
i q 21 ei ; q 21 ei q; e i t q 12 e i ; q 21 e i q; ei t
2 1
q
q
1
X
q; q)n Pn(x; q) tn; x = cos :
= ( (q;
q;
q
;
q)n q 12 n
n=0
2 1
!
q 21 ei ; q 21 ei q; e i t q 12 e i ; q 21 e
2 1
q
q
1
X
= Pn (12xn; q) tn ; x = cos :
n=0 q
!
!
q 21 ei ; q 21 ei q2 ; e i t q 23 e i ; q 23 e i q2 ; ei t
2 1
q
q3
1
2
2
X
= (( qq3 ;; qq2))n Pn(12xn; q) tn ; x = cos :
n q
n=0
71
(3.10.31)
(3.10.32)
!
!
q 21 ei ; q 23 ei q2 ; e i t q 23 e i ; q 21 e i q2 ; ei t
2 1
q2 q2
1
2
X
= (( qq2;;qq2))n Pn(12xn; q) tn ; x = cos :
n q
n=0
2 1
(3.10.33)
Remarks. The continuous q-Legendre polynomials can also be written as :
Pn(x; q) = q 12 n
n
X
(q; q2)k (q; q2)n k ei(n 2k) ; x = cos :
2
2
2
2
k=0 (q ; q )k (q ; q )n k
They are related to the continuous q-ultraspherical (or Rogers) polynomials given by (3.10.15)
in the following way :
Pn(x; q) = q 12 n Cn(x; qjq2):
References. [157], [160], [163].
3.11 Big q-Laguerre
Denition.
Orthogonality.
n; 0; x q
Pn (x; a; b; q) = 3 2 aq; bq q; q
n
aqx
1
= (b 1 q n; q) 2 1 q ;aq
n
Zaq
bq
(3.11.1)
1
q; xb :
(a 1x; b 1x; q)1 P (x; a; b; q)P (x; a; b; q)d x
m
n
q
(x; q)1
1
1
(q; q)n ( abq2 )n q(n2 ) :
= aq(1 q) (q; a(aq;b;bqab; q)q; q)1 (aq;
mn
bq; q)
1
Recurrence relation.
n
(3.11.2)
(x 1)Pn(x; a; b; q) = An Pn+1(x; a; b; q) (An + Cn) Pn(x; a; b; q) + Cn Pn 1(x; a; b; q); (3.11.3)
where
8
n+1 )(1 bqn+1 )
< An = (1 aq
:
Cn = abqn+1 (1 qn ):
q-Dierence equation.
q n(1 qn )x2y(x) = B (x)y(qx) [B (x) + D(x)] y(x) + D(x)y(q 1 x);
where
y(x) = Pn(x; a; b; q)
and
8
< B (x) = abq(1 x)
:
D(x) = (x aq)(x bq):
72
(3.11.4)
Generating functions.
(bqt; q)1 21
1 (bq; q)
aqx 1; 0 q; xt = X
n P (x; a; b; q)tn:
n
aq (
q
;
q
)
n
n=0
(3.11.5)
1 ( 1)n q(n2 )
X
0
;
0
;
x
Pn (x; a; b; q)tn:
(3.11.6)
(t; q)1 32 aq; bq q; t =
(
q
;
q
)
n
n=0
Remark. The big q-Laguerre polynomials dened by (3.11.1) and the ane q-Krawtchouk
polynomials given by (3.16.1) are related in the following way :
KnAff (q x ; p; N ; q) = Pn(q x ; p; q
N
1
; q):
References. [10], [23].
3.12 Little q-Jacobi
Denition.
pn(x; a; bjq) = 2 1
Orthogonality.
q n ; abqn+1 q; qx :
aq (3.12.1)
1
X
(bq; q)k (aq)k p (qk ; a; bjq)p (qk ; a; bjq)
m
n
k=0 (q; q)k
2
; q)1 (1 abq)(aq)n (q; bq; q)n ; 0 < aq < 1; b < q 1:
= (abq
(aq; q)1 (1 abq2n+1) (aq; abq; q)n mn
Recurrence relation.
xpn(x; a; bjq) = Anpn (x; a; bjq) (An + Cn ) pn(x; a; bjq) + Cnpn (x; a; bjq);
+1
where
8
>
>
>
>
<
>
>
>
>
:
1
aqn+1 )(1
An = qn (1(1 abq
2n+1 )(1
n)(1
Cn = aqn (1 (1abq2qn)(1
(3.12.2)
(3.12.3)
abqn+1)
abq2n+2)
bqn ) :
abq2n+1)
q-Dierence equation.
q n (1 qn)(1 abqn+1)xy(x)
= B (x)y(qx) [B (x) + D(x)] y(x) + D(x)y(q 1 x); y(x) = pn (x; a; bjq); (3.12.4)
where
8
< B (x) = a(bqx 1)
:
Generating function.
2
1
D(x) = x 1:
1 ( 1)n q(n2 )
0; 0 q; xt bqx q; t = X
pn(x; a; bjq)tn:
1 1
(
bq;
q
;
q
)
bq aq n
n=0
73
(3.12.5)
Remarks. The little q-Jacobi polynomials dened by (3.12.1) and the big q-Jacobi polynomials
given by (3.5.1) are related in the following way :
bq; q)n ( 1)n b n q n (n2 ) P (bqx; b; a; 0; q):
pn(x; a; bjq) = ((aq
n
; q )n
The little q-Jacobi polynomials and the q-Meixner polynomials dened by (3.13.1) are related
in the following way :
Mn (q x ; b; c; q) = pn( c 1 qn ; b; b 1q n x 1jq):
References. [8], [10], [18], [19], [24], [25], [31], [111], [114], [121], [127], [134], [140], [145],
[158], [160], [161], [163], [172], [176], [180], [197], [212], [214].
Special case
3.12.1 Little q-Legendre
Denition. The little q-Legendre polynomials are little q-Jacobi polynomials with a = b = 1 :
Orthogonality.
n n+1 pn (xjq) = 2 1 q ;qq q; qx :
(3.12.6)
n
qk pm (qk jq)pn(qk jq) = (1 qq2n+1) mn :
k=0
(3.12.7)
1
X
Recurrence relation.
xpn(xjq) = An pn (xjq) (An + Cn) pn(xjq) + Cnpn (xjq);
+1
where
1
8
>
>
>
>
<
qn+1 )
An = qn (1 + q(1
n+1 )(1 q2n+1 )
>
>
>
>
:
qn )
Cn = qn (1 + q(1
n)(1 q2n+1) :
q-Dierence equation.
q n(1 qn )(1 qn+1 )xy(x) = B (x)y(qx) [B (x) + D(x)] y(x) + D(x)y(q 1 x);
where
y(x) = pn (xjq)
and
8
< B (x) = qx 1
:
Generating function.
(3.12.8)
(3.12.9)
D(x) = x 1:
X
1 ( 1)n q(n2 )
qx
0
;
0
n
2 1
q q; xt 1 1 q q; t = n=0 (q; q; q)n pn (xjq)t :
References. [160], [161], [201], [221].
74
(3.12.10)
3.13
q
-Meixner
Denition.
Orthogonality.
n
Mn (q x ; b; c; q) = 21 q bq; q
x n+1
q; q c :
(3.13.1)
1
X
(bq; q)x cx q(x2) M (q x ; b; c; q)M (q x ; b; c; q)
m
n
x=0 (q; bcq; q)x
c; q)1 (q; c 1 q; q)n q n ; 0 < bq < 1; c > 0:
= ( ( bcq
mn
; q)1 (bq; q)n
(3.13.2)
Recurrence relation.
q2n+1 (1 q x )Mn (q x ) = c(1 bqn+1 )Mn+1 (q x ) +
c(1 bqn+1 ) + q(1 qn )(c + qn ) Mn (q x ) + q(1 qn )(c + qn )Mn 1(q x );
where
(3.13.3)
Mn (q x ) := Mn (q x ; b; c; q):
q-Dierence equation.
(1 qn )y(x) = B (x)y(x + 1) [B (x) + D(x)] y(x) + D(x)y(x 1);
where
y(x) = Mn (q x ; b; c; q)
and
8
x
x+1 )
< B (x) = cq (1 bq
:
Generating functions.
(3.13.4)
D(x) = (1 qx )(1 + bcqx):
1 M (q x ; b; c; q)
1 q x q; qt = X
n
tn :
1 1
bq c
(t; q)1
(
q
;
q
)
n
n=0
(3.13.5)
1
(bq; q)n M (q x ; b; c; q)tn:
b 1c 1 q x q; bqt = X
1 (3.13.6)
1 1
n
1
1
c q
(t; q)1
n=0 ( c q; q; q)n
Remarks. The q-Meixner polynomials dened by (3.13.1) and the little q-Jacobi polynomials
given by (3.12.1) are related in the following way :
Mn (q x ; b; c; q) = pn( c 1 qn ; b; b 1q n x 1jq):
The q-Meixner polynomials and the quantum q-Krawtchouk polynomials dened by (3.14.1)
are related in the following way :
Knqtm (q x ; p; N ; q) = Mn(q x ; q N 1 ; p 1; q):
References. [10], [22], [23], [114], [121], [180].
75
3.14 Quantum q-Krawtchouk
Denition.
n
Knqtm (q x ; p; N ; q) = 2 ~1 q q ; Nq
x q; pqn+1 ; n = 0; 1; 2; : ::; N:
(3.14.1)
Orthogonality.
N
X
(pq; q)N x ( 1)N x q(x2) K qtm (q x ; p; N ; q)K qtm(q x ; p; N ; q)
m
n
x=0 (q; q)x (q; q)N x
n N
N +1
n+1
= ( 1) p ((qq;; qq);Nq) n (q; pq; q)n q( 2 ) ( 2 )+Nn mn :
(3.14.2)
N
Recurrence relation.
x) +
pq2n+1(1 q x )Knqtm (q x ) = (1 qn N )Knqtm
+1 (q
(1 qn N ) + q(1 qn )(1 pqn) Knqtm (q x ) + q(1 qn )(1 pqn)Knqtm1 (q x ); (3.14.3)
where
Knqtm (q x ) := Knqtm (q x ; p; N ; q):
q-Dierence equation.
p(1 qn )y(x) = B (x)y(x + 1) [B (x) + D(x)] y(x) + D(x)y(x 1);
where
y(x) = Knqtm (q x ; p; N ; q)
and
8
x
x N)
< B (x) = q (1 q
D(x) = (1 qx )(p qx
:
Generating function.
N
1
(3.14.4)
):
N (q N ; q)
(q x t; q)1 qx N ; 0 q; q xt ' X
n K qtm (q x ; p; N ; q)tn:
pq (t; q)1 2 1
(
pq;
q
;
q
)
n n
n=0
(3.14.5)
Remarks. The quantum q-Krawtchouk polynomials dened by (3.14.1) and the q-Meixner
polynomials given by (3.13.1) are related in the following way :
Knqtm (q x ; p; N ; q) = Mn(q x ; q N 1 ; p 1; q):
The quantum q-Krawtchouk polynomials are related to the ane q-Krawtchouk polynomials
dened by (3.16.1) by the transformation q $ q 1 in the following way :
n
n
p
qtm
x
1
1
Kn (q ; p; N ; q ) = (p q; q)n q q ( 2 ) KnAff (qx N ; p 1; N ; q):
For x = 0; 1; 2; : ::; N the generating function (3.14.5) can also be written as :
(q
x t; q)x 2 1
N (q N ; q)
qx N ; 0 q; q xt = X
n K qtm (q x ; p; N ; q)tn:
pq (
pq;
q
;
q
)
n n
n=0
References. [114], [158], [160].
76
3.15
q
-Krawtchouk
Denition.
n
x
n (3.15.1)
Kn (q x ; p; N ; q) = 3~2 q ;qq N ;; 0 pq q; q
n; q x x N ; q)n
q
(
q
n
+N +1
; n = 0; 1; 2; : ::; N:
= (q N ; q) qnx 2 1 qN x n+1 q; pq
n
Orthogonality.
N
X
(q N ; q)x ( p) x K (q x ; p; N ; q)K (q x ; p; N ; q)
m
n
x=0 (q; q)x
pqN +1 ; q)n (1 + p) = (q;
( p; q N ; q)n (1 + pq2n)
N +1
( pq; q)N p N q ( 2 ) pq N n qn2 mn :
(3.15.2)
Recurrence relation.
1 q x Kn (q x ) = An Kn+1 (q x ) (An + Cn) Kn (q x ) + CnKn 1 (q x );
where
(3.15.3)
Kn (q x ) := Kn(q x ; p; N ; q)
and
8
>
>
>
>
<
n N
n)
An = (1(1+ pqq2n)(1)(1+ +pqpq
2n+1 )
>
>
>
>
:
Cn = pq2n
N
1
(1 + pqn+N )(1 qn) :
(1 + pq2n 1)(1 + pq2n)
q-Dierence equation.
q n(1 qn )(1 + pqn )y(x) = (1 qx N )y(x + 1) +
(1 qx N ) p(1 qx ) y(x) p(1 qx )y(x 1);
where
y(x) = Kn(q x ; p; N ; q):
(3.15.4)
Generating functions.
x N
N (q N ; q)
q ; 0 q; q xt q x q; pqt X
n q (n2 ) K (q x ; p; N ; q)tn: (3.15.5)
1 1
2 0
n
0
(q; q)
n
n=0
x N
~ q N; 0 q; q x t 0 1
2 1
q
pq
N +1 q;
pqN +1 x t
'
N
X
Kn (q x ; p; N ; q) tn :
N +1 ; q; q)n
n=0 ( pq
(3.15.6)
Remarks. The q-Krawtchouk polynomials dened by (3.15.1) and the dual q-Krawtchouk
polynomials given by (3.17.1) are related in the following way :
Kn(q x ; p; N ; q) = Kx ((n); pqN ; N jq)
with
(n) = q n pqn
77
or
Kn ((x); c; N jq) = Kx (q n ; cq N ; N ; q)
with
(x) = q x + cqx N :
The generating function (3.15.5) must be seen as an equality in terms of formal power series.
For x = 0; 1; 2; : ::; N this generating function can also be written as :
x N
N (q N ; q)
q ; 0 q; q xt q x q; pqt = X
n (n2 )
x
n
1 1
2 0
0 (q; q) q Kn (q ; p; N ; q)t :
n
n=0
References. [43], [114], [180], [181], [216], [217].
3.16 Ane q-Krawtchouk
Denition.
n
x KnAff (q x ; p; N ; q) = 3 ~2 q pq;; 0q; qN q; q
n
)n q( 2 ) ~ q n ; qx
= ( (pq
2 1
q N
pq; q)n
(3.16.1)
N q; q p
x
; n = 0; 1; 2; : : :; N:
Orthogonality.
N
X
(pq; q)x(q; q)N (pq) x K Aff (q x ; p; N ; q)K Aff (q x ; p; N ; q)
m
n
(
x=0 q; q)x (q; q)N x
; q)n (q; q)N n ; 0 < pq < 1:
= (pq)n N ((qpq
; q)n(q; q)N mn
(3.16.2)
Recurrence relation.
x) +
(1 q x )KnAff (q x ) = (1 qn N )(1 pqn+1)KnAff
+1 (q
(1 qn N )(1 pqn+1 ) pqn N (1 qn ) KnAff (q x ) pqn N (1 qn)KnAff1 (q x ); (3.16.3)
where
KnAff (q x ) := KnAff (q x ; p; N ; q):
q-Dierence equation.
q n(1 qn )y(x) = B (x)y(x + 1) [B (x) + D(x)] y(x) + D(x)y(x 1);
where
y(x) = KnAff (q x ; p; N ; q)
and
8
x N )(1 pqx+1)
< B (x) = (1 q
:
Generating functions.
(q
(q
(3.16.4)
D(x) = p(1 qx )qx N :
x
N (q N ; q)
N t; q)1
q q; pqt ' X
n K Aff (q x ; p; N ; q)tn:
1
1
n
x t; q)1
pq (
q
;
q
)
n
n=0
78
(3.16.5)
x N
X
N (pq; q)
q
;
0
n K Aff (q x ; p; N ; q)tn:
x
~
(3.16.6)
(pqt; q)1 2 1 q N q; q t '
n
(
q
;
q
)
n
n=0
Remarks. The ane q-Krawtchouk polynomials dened by (3.16.1) and the big q-Laguerre
polynomials given by (3.11.1) are related in the following way :
KnAff (q x ; p; N ; q) = Pn(q x ; p; q N 1; q):
The ane q-Krawtchouk polynomials are related to the quantum q-Krawtchouk polynomials
dened by (3.14.1) by the transformation q $ q 1 in the following way :
KnAff (qx ; p; N ; q 1) = (p 11q; q) Knqtm (qx N ; p 1; N ; q):
n
For x = 0; 1; 2; : ::; N the generating function (3.16.5) can also be written as :
x
X
N
N
(q N t; q)N x 1 1 qpq q; pqt = (q(q; q;)q)n KnAff (q x ; p; N ; q)tn:
n
n=0
References. [85], [86], [88], [114], [217].
3.17 Dual q-Krawtchouk
Denition.
n
x ; cqx N q
;
q
~
(3.17.1)
Kn ((x); c; N jq) = 3 2
q N ; 0 q; q
x N
n x = (q(q N ; q); qq)nnx 21 qqN x; qn+1 q; cqx+1 ; n = 0; 1; 2; : ::; N;
n
where
(x) := q x + cqx N :
Orthogonality.
N
X
(cq N ; q N ; q)x (1 cq2x N ) c x qx(2N x) K ((x))K ((x))
m
n
(1 cq N )
x=0 (q; cq; q)x
= (c 1 ; q)N (q(qN; q;)qn) (cq N )n mn ;
n
where
Recurrence relation.
(1 q x )(1
(1 qn
where
(3.17.2)
Kn((x)) := Kn ((x); c; N jq):
cqx N )Kn ((x)) =(1 qn N )Kn+1 ((x)) +
N ) + cq N (1 qn) Kn((x)) + cq N (1 qn)Kn 1 ((x));
(3.17.3)
Kn((x)) := Kn ((x); c; N jq):
q-Dierence equation.
q n(1 qn )y(x) = B (x)y(x + 1) [B (x) + D(x)] y(x) + D(x)y(x 1);
79
(3.17.4)
where
y(x) = Kn((x); c; N jq)
and
8
>
>
>
>
<
>
>
>
>
:
x N )(1 cqx N )
B (x) = (1 (1 cq2qx N )(1
cq2x N +1 )
x
cqx )
D(x) = cq2x 2N 1 (1 cq(12x Nq )(1
1 )(1
cq2x N ) :
Generating functions.
(q
(q
N t; cq N t; q)1
x t; cqx N t; q)1
'
N
X
(q N ; q)n K ((x); c; N jq)tn:
n
n=0 (q; q)n
(3.17.5)
N K ((x); c; N jq)
1
q x ; c 1q x q; cqx N t ' X
n
~
tn :
(3.17.6)
1
2
q N (q x t; q)1
(
q
;
q
)
n
n=0
Remark. The dual q-Krawtchouk polynomials dened by (3.17.1) and the q-Krawtchouk
polynomials given by (3.15.1) are related in the following way :
Kn(q x ; p; N ; q) = Kx ((n); pqN ; N jq)
with
(n) = q n pqn
or
Kn ((x); c; N jq) = Kx (q n ; cq N ; N ; q)
with
(x) = q x + cqx N :
For x = 0; 1; 2; : ::; N the generating function (3.17.5) can also be written as :
N
N
X
(q N t; q)N x (cq N t; q)x = (q(q; q;)q)n Kn ((x); c; N jq)tn:
n
n=0
References. [160], [163].
3.18 Continuous big q-Hermite
Denition.
q n; aei ; ae i q; q
Hn(x; ajq) = a
0; 0
n
i = ein 2 0 q ; ae q; qne 2i ; x = cos :
n3 2
(3.18.1)
Orthogonality. When a is real and jaj < 1, then we have the following orthogonality relation
1 Z pw(x) H (x; ajq)H (x; ajq)dx = mn ;
m
n
2
2
(qn+1 ; q)1
1
x
1
1
where
(
(
e2i ; q)1 2 = h(x; 1)h(x; 1)h(x; q 21 )h(x; q 21 ) ;
w(x) := w(x; ajq) = ae
i ; q)1 h(x; a)
80
(3.18.2)
with
h(x; ) :=
1 Y
k=0
1 2xqk + 2q2k = ei ; e i ; q 1 ; x = cos :
If a > 1, then we have another orthogonality relation given by :
1 Z pw(x) H (x; ajq)H (x; ajq)dx +
n
2
1 x2 m
1
1
X
+
1
k
<aqk a
where w(x) is as before,
and
mn ;
wk Hm (xk ; ajq)Hn(xk ; ajq) = (qn+1
; q)
1
k
k
xk = aq + 2aq
2 2k
2
2
wk = (a(q; q; )q)1 (1 (1 a aq 2)()(qa; q;) q)k q
1
k
(3.18.3)
1
3 2
2 k 12 k
1 k:
a4
Recurrence relation.
2xHn(x; ajq) = Hn (x; ajq) + aqnHn(x; ajq) + (1 qn )Hn (x; ajq):
q-Dierence equations.
+1
1
(3.18.4)
h
i
(1 q)2 Dq w~ (x; aq 21 jq)Dq y(x) + 4q n+1(1 qn)w~(x; ajq)y(x) = 0; y(x) = Hn(x; ajq); (3.18.5)
where
and
If we dene
w~(x; ajq) := wp(x; ajq2)
1 x
Dq f (x) := qf (xx) with q f (ei ) = f (q 12 ei ) f (q 21 ei ); x = cos :
q
n ; az; az 1 q
q; q
Pn(z ) := a
0; 0
then the q-dierence equation can also be written in the form
q n (1 qn )Pn(z ) = A(z )Pn (qz ) A(z ) + A(z 1) Pn(z ) + A(z 1 )Pn (q 1z );
where
A(z ) = (1 (1z 2)(1az ) qz 2 ) :
n3 2
(3.18.6)
Generating function.
References.
1 H (x; ajq)
(at; q)1 = X
n
n
(ei t; e i t; q)1 n=0 (q; q)n t ; x = cos :
81
(3.18.7)
3.19 Continuous q-Laguerre
Denitions. We have two kinds of continuous q-Laguerre polynomials coming from the continuous q-Jacobi polynomials dened by (3.10.1) and (3.10.2) :
!
+1
n 21 + 41 i 21 + 41 e i = (q(q; q;)q)n 32 q ; q qe+1;; q0
(3.19.1)
q; q
n
!
12 + 34 i
n ; q 21 + 14 ei q
e
;
q
)
1
1
1
(
q
n
(1
+
)
n
in
+
i
q 2 4 e 2 1 q 21 + 41 nei q; q 2 4 e
; x = cos =
(q; q)n
Pn (xjq)
( )
and
!
n ; q 21 ei ; q 21 e i +1 ; q)n
q
(
q
()
Pn (x; q) = (q; q) 3 2
(3.19.2)
q+1 ; q q; q ; x = cos :
n
These two q-analogues of the Laguerre polynomials are connected by the following quadratic
transformation :
Pn()(xjq2) = qnPn()(x; q):
Orthogonality. For 12 the orthogonality relations are respectively
1
(q+1 ; q)n q(+ 12 )n ;
1 Z pw(xjq) P ()(xjq)P ()(xjq)dx =
mn
m
n
+1
2
(q; q ; q)1 (q; q)n
1 x2
1
1
where
(3.19.3)
2
1
2i
2 (ei ; ei ; q 2 )1 (
e
;
q
)
1
w(xjq) := w(x; q jq) =
=
(q 12 + 41 ei ; q 21 + 43 ei ; q)1 (q 12 + 41 ei ; q 21 )1 1
21
2
= h(x; 1)h(x;21 +1)41h(x; q )12h(+x;34 q ) ;
)h(x; q
)
h(x; q
with
1 Y
h(x; ) :=
k=0
and
1 2xqk + 2q2k = ei ; e i ; q 1 ; x = cos 1 Z pw(x; q) P ()(x; q)P ()(x; q)dx
n
2
1 x2 m
1
+1
+1
= (q; q; q+11; q+1; q) (q (q;; qq; q) ; q)n qnmn ;
1
n
1
where
(3.19.4)
i
2
i
2
(e2i ; q)1
= (e ; 1 e ; q)1 = h(x; 1)h(x;1 1) ;
1
1
1
+ 2 i 2 i
(q e ; q e ; q 2 ei ; q)1 (q+ 2 ei ; q)1 h(x; q+ 2 )
w(x; q) := w(x; q; q) = with
h(x; ) :=
1 Y
k=0
1 2xqk + 2q2k = ei ; e i ; q 1 ; x = cos :
82
Recurrence relations.
2xPn (xjq) = q
( )
1 1
)
2 4 (1 qn+1 )Pn(+1
(xjq) +
1
1 ()
+
n
+1
+ q 2 4 (1 + q 2 )Pn (xjq) + q 12 + 14 (1
qn+)Pn()1(xjq): (3.19.5)
2xPn()(x; q) = q 21 (1 q2n+2)Pn(+1) (x; q) +
+ q2n++ 21 (1 + q)Pn()(x; q) + q 21 (1 q2n+2)Pn()1(x; q): (3.19.6)
q-Dierence equations.
h
i
(1 q)2Dq w~(x; q+ 12 jq)Dq y(x) +4q n+1(1 qn)w~(x; qjq)y(x) = 0; y(x) = Pn()(xjq); (3.19.7)
where
and
w~(x; qjq) := wp(x; q jq2 )
1 x
Dq f (x) := qf (xx) with q f (ei ) = f (q 12 ei ) f (q 21 ei ); x = cos :
q
i
h
(1 q)2 Dq w~(x; q+ 21 ; q)Dq y(x) +4q n+1 (1 qn )w~ (x; q; q)y(x) = 0; y(x) = Pn()(x; q); (3.19.8)
where
and
w~(x; q; q) := wp(x; q ; 2q)
1 x
Dq f (x) := qf (xx) with q f (ei ) = f (q 12 ei ) f (q 21 ei ); x = cos :
q
Generating functions.
(q+ 21 t; q+1t; q)1
12 + 14 i 21 + 41 i
(q
e t; q
e t; q)1
=
1
X
n=0
Pn()(xjq)tn; x = cos :
(3.19.9)
!
1
q 12 + 14 ei ; q 21 + 43 ei q; e i t = X
Pn()(xjq)tn ; x = cos : (3.19.10)
1
2
1
( 1 + 1 )n
+1
(ei t; q)1
q+1
n=0 (q ; q)nq 2 4
!
1
(q+1 t; q)1 q 12 ei ; q 21 ei q; q 21 e i t = X
Pn()(x; q)tn; x = cos :
q
(q 12 ei t; q)1 2 1
n=0
(3.19.11)
!
1 ( q; q) P ()(x; q)
( q 21 t; q)1 q 21 ei ; q+ 12 ei q; e i t = X
n n
n
+1 ; q)n q 12 n t ; x = cos : (3.19.12)
q+1 (ei t; q)1 2 1
(
q
n=0
References. [45].
!
q 12 e i ; q+ 12 e i q; ei t
q+1
1
X
; q)n Pn()(x; q) tn; x = cos :
= ( (qq+1
; q)n q 12 n
n=0
(q 12 t; q)1 (e i t; q)1 2 1
83
(3.19.13)
3.20 Little q-Laguerre / Wall
Denition.
Orthogonality.
n
pn(x; ajq) = 21 q aq; 0 q; qx
n
= (a 1q 1 n ; q) 2 0 q ; x
n
(3.20.1)
1
q; xa :
1
X
(aq)k p (qk ; ajq)p (qk ; ajq) = (aq)n (q; q)n ; 0 < aq < 1:
m
n
(aq; q)1 (aq; q)n mn
k=0 (q; q)k
Recurrence relation.
xpn (x; ajq) = An pn (x; ajq) (An + Cn ) pn(x; ajq) + Cnpn (x; ajq);
+1
1
where
8
<
An = qn (1 aqn+1)
:
Cn = aqn(1 qn ):
(3.20.2)
(3.20.3)
q-Dierence equation.
q n(1 qn )xy(x) = ay(qx) + (x a 1)y(x) + (1 x)y(q 1 x); y(x) = pn (x; ajq): (3.20.4)
Generating function.
X
1 ( 1)n q(n2 )
0
;
0
pn(x; ajq)tn:
(3.20.5)
(t; q)1 2 1 aq q; xt =
(
q
;
q
)
n
n=0
Remark. If we set a = q and change q to q 1 we nd the q-Laguerre polynomials dened
by (3.21.1) in the following way :
; q)n L() ( x; q):
pn (x; q jq 1) = (q(q+1
; q)n n
References. [10], [21], [75], [77], [78], [114], [160], [161], [180], [221], [224].
3.21
q
-Laguerre
Denition.
Ln
( )
+1
n
(x; q) = (q(q; q;)q)n 1 1 qq+1 q; xqn++1
n n ; x 1
q
q; qn++1 :
= (q; q) 21
0 (3.21.1)
n
Orthogonality. The q-Laguerre polynomials satisfy two kinds of orthogonality relations,
an absolutely continuous one and a discrete one. These orthogonality relations are given by,
respectively :
Z1
0
x L() (x; q)L()(x; q)dx = (q ; q)1 (q+1 ; q)n ( ) ( + 1) ; > 1 (3.21.2)
mn
n
( x; q)1 m
(q; q)1 (q; q)nqn
84
and
1
X
qk+k L()(cqk ; q)L()(cqk ; q)
m
n
k
k= 1 ( cq ; q)1
+1 ; c 1q ; q)1 (q+1 ; q)n
= (q;(qcq
+1 ;
c; c 1q; q)1 (q; q)nqn mn ; > 1; c > 0:
(3.21.3)
Recurrence relation.
q2n++1 xLn()(x; q) = (1 qn+1 )L(n+1) (x; q) +
(1 qn+1 ) + q(1 qn+ ) L(n) (x; q) + q(1 qn+ )L(n)1(x; q): (3.21.4)
q-Dierence equation.
q (1 qn)xy(x) = q (1 + x)y(qx) [1 + q (1 + x)] y(x) + y(q 1 x);
where
y(x) = L(n) (x; q):
(3.21.5)
Generating functions.
1 L() (x; q)
( xt; q)1 0; 0 q; xt = X
n
n
2 1
+1 ; q)n t :
q+1 (t; q)1
(
q
n=0
(3.21.6)
1
1 x q; q+1t = X
L(n) (x; q)tn:
(3.21.7)
(t; q)1 1 1 0 n=0
Remarks. The q-Laguerre polynomials are sometimes called the generalized Stieltjes-Wigert
polynomials.
If we change q to q 1 we obtain the little q-Laguerre (or Wall) polynomials given by (3.20.1)
in the following way :
+1 )n
pn ( x; qjq):
L(n) (x; q 1) = ((qq; q) ;qqn
n
The q-Laguerre polynomials dened by (3.21.1) and the alternative q-Charlier polynomials
given by (3.22.1) are related in the following way :
Kn (qx ; a; q) = L(x n)(aqn ; q):
n
(q; q)n
The q-Laguerre polynomials dened by (3.21.1) and the q-Charlier polynomials given by
(3.23.1) are related in the following way :
Cn( x; q ; q) = L() (x; q):
n
(q; q)n
Since the Stieltjes and Hamburger moment problems corresponding to the q-Laguerre polynomials are indeterminate there exist many dierent weight functions.
References. [8], [10], [30], [31], [45], [75], [77], [78], [96], [114], [177].
3.22 Alternative q-Charlier
85
Denition.
n
n
Kn (x; a; q) = 2 1 q ;0 aq q; qx
n = (xq1 n; q)n 1 1 xqq 1 n q; axqn+1
n
1 n
1
= ( axqn )n 2 1 q 0; x q; q a :
(3.22.1)
Orthogonality.
1
X
n (n+1
2 )
ak q(k+1
2 ) Km (qk ; a; q)Kn(qk ; a; q) = (q; q)n( aqn ; q)1 a q
(1 + aq2n) mn ; a > 0: (3.22.2)
k=0 (q; q)k
Recurrence relation.
xKn (x; a; q) = An Kn+1 (x; a; q) (An + Cn)Kn (x; a; q) + CnKn 1(x; a; q);
where
8
>
>
>
>
<
n)
An = qn (1 + aq(12n+)(1aq+ aq
2n+1 )
>
>
>
>
:
qn )
Cn = aq2n 1 (1 + aq2(1n 1)(1
+ aq2n) :
q-Dierence equation.
q n (1 qn )(1 + aqn)xy(x) = axy(qx) (ax + 1 x)y(x) + (1 x)y(q 1 x);
where
y(x) = Kn (x; a; q):
(3.22.3)
(3.22.4)
Generating functions.
2 0
1 K (x; a; q)
x 1; 0 q; xt q; aqxt X
n
n
0
1
0
(q; q) t :
n=0
n
(3.22.5)
1 ( 1)n q(n2 )
xt q; aqxt = X
(t; q)1 Kn(x; a; q)tn:
(3.22.6)
(xt; q)1 1 3 0; 0; t (
q
;
q
)
n
n=0
Remarks. The alternative q-Charlier polynomials dened by (3.22.1) and the q-Laguerre
polynomials given by (3.21.1) related in the following way :
Kn (qx ; a; q) = L(x n)(aqn ; q):
n
(q; q)n
The generating function (3.22.5) must be seen as an equality in terms of formal power series.
For x = 0; 1; 2; : ::; N this generating function can also be written as :
1 K (qx ; a; q)
X
q x ; 0 q; qxt n
x
+1
q
;
aq
t
=
tn:
0 1
2 0
0
(
q
;
q
)
n
n=0
References.
86
3.23
q
-Charlier
Denition.
Cn(q
x ; a; q )
n x
= 2 1 q 0; q q;
1
= ( a q; q)n 11
Orthogonality.
qn+1 a q n q; qn+1
a 1q a
(3.23.1)
x
:
1
X
ax q(x2) C (q x ; a; q)C (q x ; a; q) = q n ( a; q) ( a 1 q; q; q) ; a > 0:
m
n
1
n mn
x=0 (q; q)x
(3.23.2)
Recurrence relation.
q2n+1(1 q x )Cn(q x ) = aCn+1(q x ) +
[a + q(1 qn )(a + qn)] Cn(q x ) + q(1 qn)(a + qn )Cn 1(q x ); (3.23.3)
where
Cn(q x ) := Cn (q x ; a; q):
q-Dierence equation.
qny(x) = aqx y(x + 1) qx (a 1)y(x) + (1 qx )y(x 1); y(x) = Cn(q x ; a; q):
(3.23.4)
Generating functions.
1 C (q x ; a; q)
1 q x q; a 1qt = X
n
tn :
1 1
0 (t; q)1
(
q
;
q
)
n
n=0
(3.23.5)
1 C (q x ; a; q)
(q x t; q)1 0; 0 q; q x t = X
n
tn:
(3.23.6)
1 q; q ; q )
(t; q)1 2 1 a 1 q (
a
n
n=0
Remark. The q-Charlier polynomials dened by (3.23.1) and the q-Laguerre polynomials
given by (3.21.1) are related in the following way :
Cn( x; q ; q) = L() (x; q):
n
(q; q)n
References. [114], [121], [180].
3.24 Al-Salam-Carlitz I
Denition.
Una
( )
(x; q) = (
a)n q(n2 ) 21
Orthogonality.
Z
q n; x
0
1
qx
q; a :
(3.24.1)
1
(qx; a 1qx; q)1Um(a) (x; q)Un(a) (x; q)dq x
n
= ( a)n (1 q)(q; q)n (q; a; a 1q; q)1q( 2 ) mn ; a < 0:
a
87
(3.24.2)
Recurrence relation.
xUn(a) (x; q) = Un(a+1) (x; q) + (a + 1)qnUn(a) (x; q) aqn 1(1 qn)Un(a)1 (x; q):
q-Dierence equation.
(1 qn )x2y(x) = aqn 1y(qx) aqn 1 + qn (1 x)(a x) y(x) +
+ qn(1 x)(a x)y(q 1 x); y(x) = Un(a) (x; q):
(3.24.3)
(3.24.4)
Generating function.
1 U (a) (x; q)
(t; q)1 (at; q)1 = X
n
tn:
(3.24.5)
(xt; q)1
(
q
;
q
)
n
n=0
Remark. The Al-Salam-Carlitz I polynomials are related to the Al-Salam-Carlitz II polynomials dened by (3.25.1) in the following way :
Un(a) (x; q 1) = Vn(a) (x; q):
References. [10], [13], [15], [75], [77], [84], [114], [125], [135].
3.25 Al-Salam-Carlitz II
Denition.
Orthogonality.
n
n
n
Vn(a) (x; q) = ( a)n q ( 2 ) 20 q ; x q; qa :
1
X
qk2 ak V (a) (q k ; q)V (a) (q k ; q) = (q; q)nan ; a > 0:
m
n
(aq; q)1 qn2 mn
k=0 (q; q)k (aq; q)k
Recurrence relation.
a) (x; q) + (a + 1)q nV (a) (x; q) + aq
xVn(a) (x; q) = Vn(+1
n
n
2 +1
(1 qn )Vn(a)1(x; q):
q-Dierence equation.
(1 qn )x2 y(x) = (1 x)(a x)y(qx) [(1 x)(a x) + aq] y(x) +
+ aqy(q 1 x); y(x) = Vn(a) (x; q):
(3.25.1)
(3.25.2)
(3.25.3)
(3.25.4)
Generating functions.
1 ( 1)n q(n2 )
(xt; q)1 = X
(a)
n
(t; q)1 (at; q)1 n=0 (q; q)n Vn (x; q)t :
(3.25.5)
1 q2(n2 )
X
x
(3.25.6)
(at; q)1 11 at q; t = (q; q) Vn(a) (x; q)tn:
n
n=0
Remark. The Al-Salam-Carlitz II polynomials are related to the Al-Salam-Carlitz I polynomials dened by (3.24.1) in the following way :
Vn(a) (x; q 1) = Un(a) (x; q):
References. [10], [13], [15], [74], [75], [77], [84], [125].
88
3.26 Continuous q-Hermite
Denition.
Hn(xjq) = ein 20
q n ; 0 q; qne
2
i
; x = cos :
(3.26.1)
Orthogonality.
1 Z pw(x) H (xjq)H (xjq)dx = mn ;
n
2
(qn+1 ; q)1
1 x2 m
1
1
where
with
(3.26.2)
w(x) = e2i ; q 1 2 = h(x; 1)h(x; 1)h(x; q 21 )h(x; q 21 );
h(x; ) :=
1 Y
k=0
1 2xqk + 2q2k = ei ; e i ; q 1 ; x = cos :
Recurrence relation.
2xHn(xjq) = Hn+1(xjq) + (1 qn )Hn 1(xjq):
q-Dierence equation.
(1 q)2 Dq [w~(x)Dq y(x)] + 4q
where
and
n+1(1
qn)w~(x)y(x) = 0; y(x) = Hn (xjq);
(3.26.3)
(3.26.4)
w~(x) := pw(x) 2
1 x
Dq f (x) := qf (xx) with q f (ei ) = f (q 12 ei ) f (q 21 ei ); x = cos :
q
Generating functions.
1 H (xjq)
X
1
n
tn; x = cos :
=
j(ei t; q)1 j2 n=0 (q; q)n
X
1 ( 1)n q(n2 )
0
i
i
Hn(xjq)tn; x = cos :
(e t; q)1 11 ei t q; e t =
(
q
;
q
)
n
n=0
Remark. The continuous q-Hermite polynomials can also be written as :
Hn(xjq) =
(3.26.5)
(3.26.6)
n
X
(q; q)n ei(n 2k); x = cos :
(
q
;
q
)
k (q; q)n k
k=0
References. [6], [10], [20], [25], [31], [32], [37], [38], [45], [59], [62], [110], [114], [128], [137],
[138], [180], [207], [208], [209].
89
3.27 Stieltjes-Wigert
Denition.
Orthogonality.
Z1
0
n
Sn (x; q) = (q;1q) 1 1 q 0 q; xqn+1 :
n
(3.27.1)
Sm (x; q)Sn(x; q) dx = ln q (q; q)1 :
( x; q)1 ( qx 1 ; q)1
qn (q; q)n mn
(3.27.2)
Recurrence relation.
q2n+1xSn (x; q) = (1 qn+1)Sn+1 (x; q) [1 + q qn+1]Sn (x; q) + qSn 1 (x; q):
q-Dierence equation.
x(1 qn )y(x) = xy(qx) (x + 1)y(x) + y(q 1 x); y(x) = Sn (x; q):
(3.27.3)
(3.27.4)
Generating functions.
1
X
1 q; qxt =
Sn (x; q)tn:
0 1
0
(t; q)1
n=0
(3.27.5)
1
X
n
(3.27.6)
(t; q)1 0 2 0; t q; qxt = ( 1)n q( 2 ) Sn (x; q)tn:
n=0
Remark. Since the Stieltjes and Hamburger moment problems corresponding to the StieltjesWigert polynomials are indeterminate there exist many dierent weight functions. For instance,
they are also orthogonal with respect to the weight function
w(x) = p exp 2 ln2 x ; x > 0; with 2 = 2 ln1 q :
References. [30], [31], [76], [77], [84], [180], [219], [220], [226].
3.28 Discrete q-Hermite I
Denition. The discrete q-Hermite I polynomials are Al-Salam-Carlitz I polynomials with
a= 1:
n
1
hn (x; q) = Un (x; q) = q(n2 ) 2 1 q 0; x q; qx
n
n+1 q2n 1 q2;
= xn2 0 q ; q
x2 :
( 1)
(3.28.1)
Orthogonality.
Z
1
1
(qx; qx; q)1hm (x; q)hn(x; q)dq x = (1 q)(q; q)n (q; 1; q; q)1q( 2 ) mn :
n
(3.28.2)
Recurrence relation.
xhn(x; q) = hn+1 (x; q) + qn 1(1 qn)hn 1(x; q):
90
(3.28.3)
q-Dierence equation.
q n+1x2 y(x) = y(qx) (1 + q)y(x) + q(1 x2)y(q 1 x); y(x) = hn(x; q):
Generating function.
(3.28.4)
1 h (x; q)
(t; q)1 ( t; q)1 = X
n
tn:
(3.28.5)
(xt; q)1
(
q
;
q
)
n
n=0
Remark. The discrete q-Hermite I polynomials are related to the discrete q-Hermite II polynomials dened by (3.29.1) in the following way :
hn(ix; q 1 ) = in ~hn (x; q):
References. [10], [13], [62], [114], [121].
3.29 Discrete q-Hermite II
Denition. The discrete q-Hermite II polynomials are Al-Salam-Carlitz II polynomials with
a= 1:
Orthogonality.
1 h
X
k= 1
where
n n
h~ n (x; q) = i n Vn( 1) (ix; q) = i nq ( 2 ) 2 0 q ; ix q;
n
n+1 q
;
q
n
q2;
= x 2 1
0
qn
q
x2 :
i
h~ m (cqk ; q)~hn (cqk ; q) + h~ m ( cqk ; q)~hn( cqk ; q) w(cqk )qk
2
2
2
2
= 2 ((qq;; cc2;q; c c2 q2q;;qq2))1 (qq;nq2)n mn ; c > 0;
1
Recurrence relation.
(3.29.1)
2
(3.29.2)
w(x) = (ix; q) (1 ix; q) = ( x21; q2) :
1
1
1
x~hn(x; q) = h~ n+1 (x; q) + q
n
2 +1
(1 qn)~hn 1(x; q):
q-Dierence equation.
(1 qn)x2 ~hn(x; q) = (1 + x2)~hn (qx; q) (1 + x2 + q)~hn(x; q) + qh~ n(q 1 x; q):
Generating functions.
(3.29.3)
(3.29.4)
1 q(n2 )
( xt; q)1 = X
n
~
(3.29.5)
( t2 ; q2)1 n=0 (q; q)n hn (x; q)t :
X
1
n 2(n2 )
(3.29.6)
( it; q)1 1 1 ixit q; it = ( (1)q; qq) h~ n (x; q)tn:
n
n=0
Remark. The discrete q-Hermite II polynomials are related to the discrete q-Hermite I polynomials dened by (3.28.1) in the following way :
~hn(x; q 1) = i n hn(ix; q):
References.
91
Chapter 4
Limit relations between basic
hypergeometric orthogonal
polynomials
4.1 Askey-Wilson ! Continuous dual q-Hahn
The continuous dual q-Hahn polynomials dened by (3.3.1) simply follow from the Askey-Wilson
polynomials given by (3.1.1) by setting d = 0 in (3.1.1) :
pn (x; a; b; c; 0jq) = pn(x; a; b; cjq):
4.2 Askey-Wilson ! Continuous q-Hahn
The continuous q-Hahn polynomials dened by (3.4.1) can be obtained from the Askey-Wilson
polynomials given by (3.1.1) by the substitutions ! + , a ! aei , b ! bei , c ! ce i and
d ! de i :
pn(cos( + ); aei ; bei; ce i ; de ijq) = pn(cos( + ); a; b; c; d; q):
4.3 Askey-Wilson ! Big q-Jacobi
The big q-Jacobi polynomials dened by (3.5.1) can be obtained from the Askey-Wilson polynomials by setting x ! 12 a 1 x, b = a 1 q, c = a 1 q and d = a 1 in
n n (x; a; b; c; djq)
p~n (x; a; b; c; djq) = a (pab;
ac; ad; q)
n
dened by (3.1.1) and then taking the limit a ! 0 :
x
q
q
a
lim
p
~
;
a;
;
;
n
a!0
2a a a q = Pn(x; ; ; ; q):
4.4 Askey-Wilson ! Continuous q-Jacobi
If we take a = q 21 + 14 , b = q 21 + 34 , c = q 21 + 41 and d = q 21 + 34 in the denition (3.1.1)
of the Askey-Wilson polynomials and change the normalization we nd the continuous q-Jacobi
92
polynomials given by (3.10.1) :
q( 21 + 14 )n pn x; q 21 + 14 ; q 21 + 43 ; q 21 + 14 ; q 21 + 43 q
= Pn(;) (xjq):
(q; q 21 (++1) ; q 21 (++2) ; q)n
In [196] M. Rahman takes a = q 12 , b = q+ 21 , c = q+ 12 and d = q 21 to obtain after a change of
normalization the continuous q-Jacobi polynomials dened by (3.10.2) :
q 12 npn x; q 21 ; q+ 21 ; q+ 12 ; q 21 q
= Pn(;)(x; q):
(q; q; q; q)n
As was pointed out in section 0.6 these two q-analogues of the Jacobi polynomials are not really
dierent, since they are connected by the quadratic transformation
q)n qn P (;)(x; q):
Pn(;)(xjq2) = ( q(+q;+1
n
; q)n
4.5 Askey-Wilson ! Continuous q-ultraspherical / Rogers
If we set a = 21 , b = 21 q 12 , c = 21 and d = 12 q 12 in the denition (3.1.1) of the Askey-Wilson
polynomials and change the normalization we obtain the continuous q-ultraspherical (or Rogers)
polynomials dened by (3.10.15). In fact we have :
( 2 ; q)npn x; 21 ; 21 q 12 ; 12 ; 21 q 21 q
= Cn(x; jq):
(q 12 ; ; q 12 ; q; q)n
4.6
q
-Racah ! Big q-Jacobi
The big q-Jacobi polynomials dened by (3.5.1) can be obtained from the q-Racah polynomials
by setting = 0 in the denition (3.2.1) :
Rn((x); a; b; c; 0jq) = Pn(q x ; a; b; c; q):
4.7
q
-Racah ! q-Hahn
The q-Hahn polynomials follow from the q-Racah polynomials by the substitution = 0 and
q = q N in the denition (3.2.1) of the q-Racah polynomials :
Rn((x); ; ; q N 1; 0jq) = Qn (q x ; ; ; N jq):
Another way to obtain the q-Hahn polynomials from the q-Racah polynomials is by setting = 0
and = 1 q N 1 in the denition (3.2.1) :
Rn((x); ; ; 0; 1q N 1 jq) = Qn (q x ; ; ; N jq):
And if we take q = q N , ! qN +1 and = 0 in the denition (3.2.1) of the q-Racah
polynomials we nd the q-Hahn polynomials given by (3.6.1) in the following way :
Rn((x); q N 1 ; qN +1 ; ; 0jq) = Qn(q x ; ; ; N jq):
Note that (x) = q x in each case.
93
4.8
q
4.9
q
-Racah ! Dual q-Hahn
To obtain the dual q-Hahn polynomials from the q-Racah polynomials we have to take = 0 and
q = q N in (3.2.1) :
Rn((x); q N 1 ; 0; ; jq) = Rn((x); ; ; N jq);
with
(x) = q x + qx+1 :
We may also take = 0 and = 1q N 1 in (3.2.1) to obtain the dual q-Hahn polynomials
from the q-Racah polynomials :
Rn((x); 0; 1q N 1; ; jq) = Rn((x); ; ; N jq);
with
(x) = q x + qx+1 :
And if we take q = q N , ! qN +1 and = 0 in the denition (3.2.1) of the q-Racah
polynomials we nd the dual q-Hahn polynomials given by (3.7.1) in the following way :
Rn((x); ; 0; q N 1; qN +1 jq) = Rn((x); ; ; N jq);
with
(x) = q x + qx+1:
-Racah ! q-Krawtchouk
The q-Krawtchouk polynomials dened by (3.15.1) can be obtained from the q-Racah polynomials by setting q = q N , = pqN and = = 0 in the denition (3.2.1) of the q-Racah
polynomials :
Rn(q x ; q N 1; pqN ; 0; 0jq) = Kn (q x ; p; N ; q):
Note that (x) = q x in this case.
4.10
q
-Racah ! Dual q-Krawtchouk
The dual q-Krawtchouk polynomials dened by (3.17.1) easily follow from the q-Racah polynomials
given by (3.2.1) by using the substitutions = = 0, q = q N and = c :
Rn ((x); 0; 0; q
Note that
N
; cjq) = Kn ((x); c; N jq):
1
(x) = (x) = q x + cqx N :
4.11 Continuous dual q-Hahn ! Al-Salam-Chihara
The Al-Salam-Chihara polynomials dened by (3.8.1) simply follow from the continuous dual
q-Hahn polynomials by taking c = 0 in the denition (3.3.1) of the continuous dual q-Hahn
polynomials :
pn(x; a; b; 0jq) = Qn(x; a; bjq):
94
4.12 Continuous q-Hahn ! q-Meixner-Pollaczek
The q-Meixner-Pollaczek polynomials dened by (3.9.1) simply follow from the continuous q-Hahn
polynomials if we set d = a and b = c = 0 in the denition (3.4.1) of the continuous q-Hahn
polynomials :
pn (cos( + ); a; 0; 0; a; q) = P (cos( + ); ajq):
n
(q; q)
n
4.13 Big q-Jacobi ! Big q-Laguerre
If we set b = 0 in the denition (3.5.1) of the big q-Jacobi polynomials we obtain the big q-Laguerre
polynomials given by (3.11.1) :
Pn(x; a; 0; c; q) = Pn(x; a; c; q):
4.14 Big q-Jacobi ! Little q-Jacobi
The little q-Jacobi polynomials dened by (3.12.1) can be obtained from the big q-Jacobi polynomials by the substitution x ! cqx in the denition (3.5.1) and then by the limit c ! 1 :
lim P (cqx; a; b; c; q) = pn(x; a; bjq):
c!1 n
4.15 Big q-Jacobi ! q-Meixner
If we take the limit a ! 1 in the denition (3.5.1) of the big q-Jacobi polynomials we simply
obtain the q-Meixner polynomials dened by (3.13.1) :
lim P (q x ; a; b; c; q) = Mn (q x ; c; b 1; q):
a!1 n
4.16
q
-Hahn ! Little q-Jacobi
If we set x ! N x in the denition (3.6.1) of the q-Hahn polynomials and take the limit N ! 1
we nd the little q-Jacobi polynomials :
lim Q (qx N ; ; ; N jq) = pn (qx ; ; jq);
N !1 n
where pn (qx ; ; jq) is dened by (3.12.1).
4.17
q
4.18
q
-Hahn ! q-Meixner
The q-Meixner polynomials dened by (3.13.1) can be obtained from the q-Hahn polynomials by
setting = b and = b 1c 1 q N 1 in the denition (3.6.1) of the q-Hahn polynomials and
letting N ! 1 :
lim Q q x ; b; b 1c 1q N 1 ; N jq = Mn (q x ; b; c; q):
N !1 n
-Hahn ! Quantum q-Krawtchouk
The quantum q-Krawtchouk polynomials dened by (3.14.1) simply follow from the q-Hahn polynomials by setting = p in the denition (3.6.1) of the q-Hahn polynomials and taking the limit
!1:
lim Q (q x ; ; p; N jq) = Knqtm (q x ; p; N ; q):
!1 n
95
4.19
q
4.20
q
-Hahn ! q-Krawtchouk
If we set = 1 q 1p in the denition (3.6.1) of the q-Hahn polynomials and then let ! 0
we obtain the q-Krawtchouk polynomials dened by (3.15.1) :
p ; N q = K (q x ; p; N ; q):
x
lim
Q
q
;
;
n
!0 n
q -Hahn ! Ane q-Krawtchouk
The ane q-Krawtchouk polynomials dened by (3.16.1) can be obtained from the q-Hahn polynomials by the substitution = p and = 0 in (3.6.1) :
Qn(q x ; p; 0; N jq) = KnAff (q x ; p; N ; q):
4.21 Dual q-Hahn ! Ane q-Krawtchouk
The ane q-Krawtchouk polynomials dened by (3.16.1) can be obtained from the dual q-Hahn
polynomials by the substitution = p and = 0 in (3.7.1) :
Rn((x); p; 0; N jq) = KnAff (q x ; p; N ; q):
Note that (x) = q x in this case.
4.22 Dual q-Hahn ! Dual q-Krawtchouk
The dual q-Krawtchouk polynomials dened by (3.17.1) can be obtained from the dual q-Hahn
polynomials by setting = c 1 q N 1 in (3.7.1) and then letting ! 0 :
c
N
1
lim
R
(
x
);
;
q
n
q = Kn ((x); c; N jq):
!0
4.23 Al-Salam-Chihara ! Continuous big q-Hermite
If we take the limit b ! 0 in the denition (3.8.1) of the Al-Salam-Chihara polynomials we simply
obtain the continuous big q-Hermite polynomials given by (3.18.1) :
lim Q (x; a; bjq) = Hn(x; ajq):
b!0 n
4.24 Al-Salam-Chihara ! Continuous q-Laguerre
The continuous q-Laguerre polynomials dened by (3.19.1)
can be obtained from the Al-SalamChihara polynomials given by (3.8.1) by taking a = q 12 + 41 and b = q 21 + 43 :
Qn x; q 21 + 14 ; q 21 + 34 q = ((12q; +q)41n)n Pn()(xjq):
q
4.25
-Meixner-Pollaczek ! Continuous q-ultraspherical /
Rogers
q
If we take = 0 and a = in the denition (3.9.1) of the q-Meixner-Pollaczek polynomials we
obtain the continuous q-ultraspherical (or Rogers) polynomials given by (3.10.15) :
Pn(cos ; jq) = Cn(cos ; jq):
96
4.26 Continuous q-Jacobi ! Continuous q-Laguerre
The continuous q-Laguerre polynomials given by (3.19.1) and (3.19.2) follow simply from the
continuous q-Jacobi polynomials dened by (3.10.1) and (3.10.2) respectively by taking the limit
!1:
lim P (;)(xjq) = Pn()(xjq)
!1 n
and
Pn()(x; q) :
(; )
lim
P
(
x
;
q
)
=
!1 n
( q; q)
n
4.27 Continuous q-ultraspherical / Rogers ! Continuous
q -Hermite
The continuous q-Hermite polynomials dened by (3.26.1) can be obtained from the continuous
q-ultraspherical (or Rogers) polynomials given by (3.10.15) by taking the limit ! 0. In fact we
have
Hn(xjq) :
lim
C
(
x
;
j
q
)
=
n
!0
(q; q)n
4.28 Big q-Laguerre ! Little q-Laguerre / Wall
The little q-Laguerre (or Wall) polynomials dened by (3.20.1) can be obtained from the big
q-Laguerre polynomials by taking x ! bqx in (3.11.1) and then letting b ! 1 :
lim P (bqx; a; b; q) = pn (x; ajq):
b!1 n
4.29 Big q-Laguerre ! Al-Salam-Carlitz I
If we set x ! aqx and b ! ab in the denition (3.11.1) of the big q-Laguerre polynomials and take
the limit a ! 0 we obtain the Al-Salam-Carlitz I polynomials given by (3.24.1) :
lim Pn(aqxa;na; ab; q) = Un(b)(x; q):
a!0
4.30 Little q-Jacobi ! Little q-Laguerre / Wall
The little q-Laguerre (or Wall) polynomials dened by (3.20.1) are little q-Jacobi polynomials with
b = 0. So if we set b = 0 in the denition (3.12.1) of the little q-Jacobi polynomials we obtain the
little q-Laguerre (or Wall) polynomials :
pn(x; a; 0jq) = pn(x; ajq):
4.31 Little q-Jacobi ! q-Laguerre
If we substitute a = q and x ! b 1 q 1 x in the denition (3.12.1) of the little q-Jacobi polynomials and then let b tend to innity we nd the q-Laguerre polynomials given by (3.21.1) :
x ; q; b q = (q; q)n L() (x; q):
lim
p
b!1 n
bq
(q+1 ; q)n n
97
4.32 Little q-Jacobi ! Alternative q-Charlier
If we set b ! a 1 q 1 b in the denition (3.12.1) of the little q-Jacobi polynomials and then take
the limit a ! 0 we obtain the alternative q-Charlier polynomials given by (3.22.1) :
b
lim
p
x
;
a;
n
a!0
aq q = Kn (x; b; q):
4.33
q
4.34
q
4.35
q
-Meixner ! q-Laguerre
The q-Laguerre polynomials dened by (3.21.1) can be obtained from the q-Meixner polynomials
given by (3.13.1) by setting b = q and q x ! cq x in the denition (3.13.1) of the q-Meixner
polynomials and then taking the limit c ! 1 :
; q)n L() (x; q):
lim M (cq x; q; c; q) = (q(q+1
c!1 n
; q)n n
-Meixner ! q-Charlier
The q-Meixner polynomials and the q-Charlier polynomials dened by (3.13.1) and (3.23.1) respectively are simply related by the limit b ! 0 in the denition (3.13.1) of the q-Meixner polynomials.
In fact we have
Mn (x; 0; a; q) = Cn(x; a; q):
-Meixner ! Al-Salam-Carlitz II
The Al-Salam-Carlitz II polynomials dened by (3.25.1) can be obtained from the q-Meixner
polynomials dened by (3.13.1) by setting b = c 1a in the denition (3.13.1) of the q-Meixner
polynomials and then taking the limit c # 0 :
a ; c; q = 1 n q(n2 ) V (a) (x; q):
x
;
lim
M
n
n
c# 0
c
a
4.36 Quantum q-Krawtchouk ! Al-Salam-Carlitz II
If we set p = a 1 q N 1 in the denition (3.14.1) of the quantum q-Krawtchouk polynomials and
let N ! 1 we obtain the Al-Salam-Carlitz II polynomials given by (3.25.1). In fact we have
1 n q(n2 ) V (a) (x; q):
qtm (x; a 1q N 1 ; N ; q) =
lim
K
n
N !1 n
a
4.37
q
4.38
q
-Krawtchouk ! Alternative q-Charlier
If we set x ! N x in the denition (3.15.1) of the q-Krawtchouk polynomials and then take the
limit N ! 1 we obtain the alternative q-Charlier polynomials dened by (3.22.1) :
x N ; p; N ; q = Kn(qx ; p; q):
lim
K
q
n
N !1
-Krawtchouk ! q-Charlier
The q-Charlier polynomials given by (3.23.1) can be obtained from the q-Krawtchouk polynomials dened by (3.15.1) by setting p = a 1 q N in the denition (3.15.1) of the q-Krawtchouk
polynomials and then taking the limit N ! 1 :
x ; a 1q N ; N ; q = Cn(q x ; a; q):
q
lim
K
n
N !1
98
4.39 Ane q-Krawtchouk ! Little q-Laguerre / Wall
If we set x ! N x in the denition (3.16.1) of the ane q-Krawtchouk polynomials and take the
limit N ! 1 we simply obtain the little q-Laguerre (or Wall) polynomials dened by (3.20.1) :
lim K Aff (qx N ; p; N ; q) = pn(qx ; p; q):
N !1 n
4.40 Dual q-Krawtchouk ! Al-Salam-Carlitz I
If we set c = a 1 in the denition (3.17.1) of the dual q-Krawtchouk polynomials and take the
limit N ! 1 we simply obtain the Al-Salam-Carlitz I polynomials given by (3.24.1) :
1 ; N q = 1 n q (n2 ) U (a) (qx ; q):
K
lim
(
x
);
n
N !1 n
a a
Note that (x) = q x + a 1 qx N .
4.41 Continuous big q-Hermite ! Continuous q-Hermite
The continuous q-Hermite polynomials dened by (3.26.1) can easily be obtained from the continuous big q-Hermite polynomials given by (3.18.1) by taking a = 0 :
Hn(x; 0jq) = Hn(xjq):
4.42 Continuous q-Laguerre ! Continuous q-Hermite
The continuous q-Hermite polynomials given by (3.26.1) can be obtained from the continuous
q-Laguerre polynomials dened by (3.19.1) by taking the limit ! 1 in the following way :
Pn()(xjq) = Hn(xjq) :
lim
!1 q( 21 + 14 )n
(q; q)n
4.43
q
-Laguerre ! Stieltjes-Wigert
If we set x ! xq in the denition (3.21.1) of the q-Laguerre polynomials and take the limit
! 1 we simply obtain the Stieltjes-Wigert polynomials given by (3.27.1) :
; q = Sn (x; q):
()
xq
lim
L
n
!1
4.44 Alternative q-Charlier ! Stieltjes-Wigert
The Stieltjes-Wigert polynomials dened by (3.27.1) can be obtained from the alternative qCharlier polynomials by setting x ! a 1x in the denition (3.22.1) of the alternative q-Charlier
polynomials and then taking the limit a ! 1. In fact we have
x
lim
K
;
a
;
q
= (q; q)n Sn (x; q):
n
a!1
a
4.45
q
-Charlier ! Stieltjes-Wigert
If we set q x ! ax in the denition (3.23.1) of the q-Charlier polynomials and take the limit
a ! 1 we obtain the Stieltjes-Wigert polynomials given by (3.27.1) in the following way :
lim C (ax; a; q) = (q; q)n Sn (x; q):
a!1 n
99
4.46 Al-Salam-Carlitz I ! Discrete q-Hermite I
The discrete q-Hermite I polynomials dened by (3.28.1) can easily be obtained from the AlSalam-Carlitz I polynomials given by (3.24.1) by the substitution a = 1 :
Un( 1) (x; q) = hn(x; q):
4.47 Al-Salam-Carlitz II ! Discrete q-Hermite II
The discrete q-Hermite II polynomials dened by (3.29.1) follow from the Al-Salam-Carlitz II
polynomials given by (3.25.1) by the substitution a = 1 in the following way :
i nVn( 1)(ix; q) = h~ n (x; q):
100
Chapter 5
From basic to classical
hypergeometric orthogonal
polynomials
5.1 Askey-Wilson ! Wilson
To nd the Wilson polynomials dened by (1.1.1) from the Askey-Wilson polynomials we set
a ! qa , b ! qb, c ! qc, d ! qd and ei = qix (or = ln qx ) in the denition (3.1.1) and take the
limit q " 1 :
pn( 12 qix + q ix ; qa ; qb; qc; qd jq) = W (x2; a; b; c; d):
lim
n
q "1
(1 q)3n
5.2
q
-Racah ! Racah
If we set ! q , ! q , ! q , ! q in the denition (3.2.1) of the q-Racah polynomials
and let q " 1 we easily obtain the Racah polynomials dened by (1.2.1) :
lim
R ((x); q; q ; q ; q jq) = Rn((x); ; ; ; );
q "1 n
where
8
<
(x) = q x + qx+ ++1
:
(x) = x(x + + + 1):
5.3 Continuous dual q-Hahn ! Continuous dual Hahn
To nd the continuous dual Hahn polynomials dened by (1.3.1) from the continuous dual q-Hahn
polynomials we set a ! qa , b ! qb , c ! qc and ei = qix (or = ln qx ) in the denition (3.3.1)
and take the limit q " 1 :
p
n( 21 qix + q ix ; qa; qb ; qcjq)
lim
= Sn (x2; a; b; c):
q "1
(1 q)2n
5.4 Continuous q-Hahn ! Continuous Hahn
If we set a ! qa , b ! qb , c ! qc , d ! qd and e i = qix (or = ln q x ) in the denition
(3.4.1) of the continuous q-Hahn polynomials and take the limit q " 1 we nd the continuous Hahn
101
polynomials given by (1.4.1) in the following way :
pn(cos(ln q x + ); qa ; qb; qc; qd ; q) = ( 2 sin )np (x; a; b; c; d):
lim
n
q "1
(1 q)n (q; q)n
5.5 Big q-Jacobi ! Jacobi
If we set c = 0, a = q and b = q in the denition (3.5.1) of the big q-Jacobi polynomials and let
q " 1 we nd the Jacobi polynomials given by (1.8.1) :
Pn(;)(2x 1) :
lim
P
(
x
;
q
;
q
;
0;
q
)
=
q "1 n
Pn(;)(1)
If we take c = q for arbitrary real instead of c = 0 we nd
(; )
; q ; q ; q) = Pn (x) :
lim
P
(
x
;
q
n
q "1
Pn(;)(1)
5.5.1 Big q-Legendre ! Legendre / Spherical
If we set c = 0 in the denition (3.5.7) of the big q-Legendre polynomials and let q " 1 we simply
obtain the Legendre (or spherical) polynomials dened by (1.8.40) :
lim
P (x; 0; q) = Pn(2x 1):
q "1 n
If we take c = q for arbitrary real instead of c = 0 we nd
lim
P (x; q ; q) = Pn(x):
q "1 n
5.6
q
-Hahn ! Hahn
The Hahn polynomials dened by (1.5.1) simply follow from the q-Hahn polynomials given by
(3.6.1), after setting ! q and ! q , in the following way :
lim
Q (q x ; q; q ; N jq) = Qn(x; ; ; N ):
q "1 n
5.7 Dual q-Hahn ! Dual Hahn
The dual Hahn polynomials given by (1.6.1) follow from the dual q-Hahn polynomials by simply
taking the limit q " 1 in the denition (3.7.1) of the dual q-Hahn polynomials after applying the
substitution ! q and ! q :
lim
R ((x); q ; q ; N jq) = Rn((x); ; ; N );
q "1 n
where
8
<
(x) = q x + qx+ ++1
:
(x) = x(x + + + 1):
5.8 Al-Salam-Chihara ! Meixner-Pollaczek
If we set a = qe i , b = qei and ei = qix ei in the denition (3.8.1) of the Al-Salam-Chihara
polynomials and take the limit q " 1 we obtain the Meixner-Pollaczek polynomials given by (1.7.1)
in the following way :
cos(ln qx + ); qei ; qe i jq
Q
n
lim
= Pn()(x; ):
q "1
(q; q)n
102
5.9
q
-Meixner-Pollaczek ! Meixner-Pollaczek
To nd the Meixner-Pollaczek polynomials dened by (1.7.1) from the q-Meixner-Pollaczek polynomials we substitute a = q and ei = q ix (or = ln q x ) in the denition (3.9.1) of the
q-Meixner-Pollaczek polynomials and take the limit q " 1 to nd :
lim
P (cos(ln q x + ); qjq) = Pn()(x; ):
q "1 n
5.10 Continuous q-Jacobi ! Jacobi
If we take the limit q " 1 in the denitions (3.10.1) and (3.10.2) of the continuous q-Jacobi
polynomials we simply nd the Jacobi polynomials dened by (1.8.1) :
lim
P (;)(xjq) = Pn(;)(x)
q "1 n
and
lim
P (;)(x; q) = Pn(;)(x):
q "1 n
5.10.1 Continuous q-ultraspherical / Rogers
spherical
! Gegenbauer / Ultra-
If we set = q in the denition (3.10.15) of the continuous q-ultraspherical (or Rogers) polynomials and let q tend to one we obtain the Gegenbauer (or ultraspherical) polynomials given by
(1.8.10) :
lim
C (x; qjq) = Cn() (x):
q "1 n
5.10.2 Continuous q-Legendre ! Legendre / Spherical
The Legendre (or spherical) polynomials dened by (1.8.40) easily follow from the continuous
q-Legendre polynomials given by (3.10.25) by taking the limit q " 1 :
lim
P (x; q) = Pn(x):
q "1 n
Of course, we also have
lim
P (xjq) = Pn(x):
q "1 n
5.11 Big q-Laguerre ! Laguerre
The Laguerre polynomials dened by (1.11.1) can be obtained from the big q-Laguerre polynomials
by the substitution a = q and b = (1 q) 1 q in the denition (3.11.1) of the big q-Laguerre
polynomials and the limit q " 1 :
()
; (1 q) 1 q ; q) = Ln (x 1) :
lim
P
(
x
;
q
n
q "1
L(n)(0)
5.12 Little q-Jacobi ! Jacobi
The Jacobi polynomials dened by (1.8.1) simply follow from the little q-Jacobi polynomials
dened by (3.12.1) in the following way :
Pn(;)(1 2x) :
lim
p
(
x
;
q
;
q
j
q
)
=
q "1 n
Pn(;)(1)
103
5.12.1 Little q-Legendre ! Legendre / Spherical
If we take the limit q " 1 in the denition (3.12.6) of the little q-Legendre polynomials we simply
nd the Legendre (or spherical) polynomials given by (1.8.40) :
lim
p (xjq) = Pn(1 2x):
q "1 n
5.12.2 Little q-Jacobi ! Laguerre
If we take a = q, b = q for arbitrary real and x ! 21 (1 q)x in the denition (3.12.1) of
the little q-Jacobi polynomials and then take the limit q " 1 we obtain the Laguerre polynomials
given by (1.11.1) :
1 (1 q)x; q; q q = L(n) (x) :
lim
p
q "1 n 2
L(n) (0)
5.13
q
-Meixner ! Meixner
To nd the Meixner polynomials dened by (1.9.1) from the q-Meixner polynomials given by
(3.13.1) we set b = q 1 and c ! (1 c) 1 c and let q " 1 :
c
x
1
lim
M q ; q ; 1 c ; q = Mn (x; ; c):
q "1 n
5.14 Quantum q-Krawtchouk ! Krawtchouk
The Krawtchouk polynomials given by (1.10.1) easily follow from the quantum q-Krawtchouk
polynomials dened by (3.14.1) in the following way :
lim
K qtm (q x ; p; N ; q) = Kn (x; p 1; N ):
q "1 n
5.15
q
-Krawtchouk ! Krawtchouk
If we take the limit q " 1 in the denition (3.15.1) of the q-Krawtchouk polynomials we simply
nd the Krawtchouk polynomials given by (1.10.1) in the following way :
1
x
lim
K (q ; p; N ; q) = Kn x; p + 1 ; N :
q "1 n
5.16 Ane q-Krawtchouk ! Krawtchouk
If we let q " 1 in the denition (3.16.1) of the ane q-Krawtchouk polynomials we obtain :
lim
K Aff (q x ; p; N jq) = Kn (x; 1 p; N );
q "1 n
where Kn (x; 1 p; N ) is the Krawtchouk polynomial dened by (1.10.1).
5.17 Dual q-Krawtchouk ! Krawtchouk
If we set c = 1 p 1 in the denition (3.17.1) of the dual q-Krawtchouk polynomials and take the
limit q " 1 we simply nd the Krawtchouk polynomials given by (1.10.1) :
1
lim
K (x); 1 p ; N jq = Kn (x; p; N ):
q "1 n
104
5.18 Continuous big q-Hermite ! Hermite
q
If we set a = 0 and x ! x 12 (1 q) in the denition (3.18.1) of the continuous big q-Hermite
polynomials and let q tend to one, we obtain the Hermite polynomials given by (1.13.1) in the
following way :
1 Hn x 1 2 q 2 ; 0 q
lim
= Hn(x):
n
1 q 2
q "1
2
q
p
If we take a ! a 2(1 q) and x ! x 12 (1 q) in the denition (3.18.1) of the continuous
big q-Hermite polynomials and take the limit q " 1 we nd the Hermite polynomials dened by
(1.13.1) with shifted argument :
lim
q "1
Hn x
1
2
q 12 ; ap2(1
1
2
q n2
q) q
= Hn(x a):
5.19 Continuous q-Laguerre ! Laguerre
If we set x ! qx in the denitions (3.19.1) and (3.19.2) of the continuous q-Laguerre polynomials
and take the limit q " 1 we nd the Laguerre polynomials dened by (1.11.1). In fact we have :
lim
P ()(qx jq) = L(n) (2x)
q "1 n
and
lim
P ()(qx ; q) = L(n) (x):
q "1 n
5.20 Little q-Laguerre / Wall ! Laguerre
If we set a = q and x ! (1 q)x in the denition (3.20.1) of the little q-Laguerre (or Wall)
polynomials and let q tend to one, we obtain the Laguerre polynomials given by (1.11.1) :
lim
p ((1 q)x; qjq) = Ln()(x) :
q "1 n
Ln (0)
( )
5.20.1 Little q-Laguerre / Wall ! Charlier
If we set a ! (q 1)a and x ! qx in the denition (3.20.1) of the little q-Laguerre (or Wall)
polynomials and take the limit q " 1 we obtain the Charlier polynomials given by (1.12.1) in the
following way :
pn(qx ; (q 1)ajq) = Cn (x; a) :
lim
q "1
(1 q)n
an
5.21
q
-Laguerre ! Laguerre
If we set x ! (1 q)x in the denition (3.21.1) of the q-Laguerre polynomials and take the limit
q " 1 we obtain the Laguerre polynomials given by (1.11.1) :
lim
L() ((1 q)x; q) = L(n)(x):
q "1 n
105
5.21.1 q-Laguerre ! Charlier
If we set x ! q x and q = a 1(q 1) 1 (or = (ln q) 1 ln(q 1)a) in the denition (3.21.1)
of the q-Laguerre polynomials, multiply by (q; q)n , and take the limit q " 1 we obtain the Charlier
polynomials given by (1.12.1) :
lim
(q; q)nLn() ( q x ; q) = Cn(x; a); q = a(q 1 1) or = ln(qln q 1)a :
q "1
5.22 Alternative q-Charlier ! Charlier
If we set x ! qx and a ! a(1 q) in the denition (3.22.1) of the alternative q-Charlier polynomials
and take the limit q " 1 we nd the Charlier polynomials given by (1.12.1) :
Kn(qx ; a(1 q); q) = anC (x; a):
lim
n
q "1
(q 1)n
5.23
q
-Charlier ! Charlier
If we set a ! a(1 q) in the denition (3.23.1) of the q-Charlier polynomials and take the limit
q " 1 we obtain the Charlier polynomials dened by (1.12.1) :
lim
C (q x ; a(1 q); q) = Cn(x; a):
q "1 n
5.24 Al-Salam-Carlitz I ! Charlier
If we set a ! a(q 1) and x ! qx in the denition (3.24.1) of the Al-Salam-Carlitz I polynomials
and take the limit q " 1 after dividing by an (1 q)n we obtain the Charlier polynomials dened
by (1.12.1) :
Un(a(q 1))(qx ; q) = an C (x; a):
lim
n
q "1
(1 q)n
5.24.1 Al-Salam-Carlitz Ip! Hermite
p
If we set x ! x 1 q2 and a !n a 1 q2 1 in the denition (3.24.1) of the Al-Salam-Carlitz I
polynomials, divide by (1 q2 ) 2 , and let q tend to one we obtain the Hermite polynomials given
by (1.13.1) with shifted argument. In fact we have
p
(a 1
U
n
lim
q "1
q2
p
(x 1 q2 ; q) = Hn(x a) :
2n
(1 q2 ) n2
1)
5.25 Al-Salam-Carlitz II ! Charlier
If we set a ! a(1 q) and x ! q x in the denition (3.25.1) of the Al-Salam-Carlitz II polynomials
and taking the limit q " 1 we nd
Vn(a(1 q)) (q x ; q) = anC (x; a):
lim
n
q "1
(q 1)n
106
5.25.1 Al-Salam-Carlitz pII ! Hermite
p
If we set x ! x 1 q2 and a ! a 1 q2 +1 in the denition (3.25.1) of the Al-Salam-Carlitz II
polynomials, divide by (1 q2 ) n2 , and let q tend to one we obtain the Hermite polynomials given
by (1.13.1) with shifted argument. In fact we have
(a
lim Vn
p1
q "1
q2 +1)
p
(x 1 q2 ; q) = Hn(x 2) :
2n
(1 q2 ) n2
5.26 Continuous q-Hermite ! Hermite
The Hermite polynomials dened by (1.13.1) q
can be obtained from the continuous q-Hermite
polynomials given by (3.26.1) by setting x ! x 12 (1 q). In fact we have
lim
q "1
Hn x
1
1
2
2
q 12 q
q n2
= Hn(x):
5.27 Stieltjes-Wigert ! Hermite
The Hermite polynomials dened by (1.13.1)
p can be obtained from the Stieltjes-Wigert polynomials
given by (3.27.1) by setting x ! q 1x 2(1 q) + 1 and taking the limit q " 1 in the following
way :
p
1
(
q
;
q
)
S
(
q
x
2(1 q) + 1; q) = ( 1)n H (x):
n
n
lim
n
n
1 q 2
q "1
2
5.28 Discrete q-Hermite I ! Hermite
The Hermite polynomials dened by (1.13.1) can be found from the discrete q-Hermite I polynomials given by (3.28.1) in the following way :
p
hn x 1 q2 ; q Hn(x)
lim
= 2n :
q "1
(1 q2) n2
5.29 Discrete q-Hermite II ! Hermite
The Hermite polynomials dened by (1.13.1) can also be found from the discrete q-Hermite II
polynomials given by (3.29.1) in a similar way :
p
h~ n x 1 q2 ; q Hn(x)
lim
= 2n :
q "1
(1 q2) n2
107
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Mathematics 15, 1963, 332-349.
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American Mathematical Society 321, 1990, 363-378.
[233] J. Zeng : Linearisation de produits de polyn^omes de Meixner, Krawtchouk, et Charlier.
SIAM Journal on Mathematical Analysis 21, 1990, 1349-1368.
120
Index
Ane q-Krawtchouk polynomials, 78, 96, 99,
104
Al-Salam-Carlitz I polynomials, 87, 97, 99,
100, 106
Al-Salam-Carlitz II polynomials, 88, 98, 100,
106, 107
Al-Salam-Chihara polynomials, 62, 94, 96,
102
Alternative q-Charlier polynomials, 85, 98,
99, 106
Askey-scheme, 22
Askey-Wilson polynomials, 50, 92, 93, 101
Big q-Jacobi polynomials, 57, 92, 93, 95, 102
Big q-Laguerre polynomials, 72, 95, 97, 103
Big q-Legendre polynomials, 58, 102
Charlier polynomials, 40, 46, 47, 105, 106
Chebyshev polynomials, 35
Continuous q-Hahn polynomials, 55, 92, 95,
101
Continuous q-Hermite polynomials, 89, 97,
99, 107
Continuous q-Jacobi polynomials, 65, 92, 97,
103
Continuous q-Laguerre polynomials, 82, 96,
97, 99, 105
Continuous q-Legendre polynomials, 70, 103
Continuous q-ultraspherical polynomials, 68,
93, 96, 97, 103
Continuous big q-Hermite polynomials, 80,
96, 99, 105
Continuous dual q-Hahn polynomials, 54, 92,
94, 101
Continuous dual Hahn polynomials, 27, 43,
44, 101
Continuous Hahn polynomials, 28, 43, 44,
101
Discrete q-Hermite I polynomials, 90, 100,
107
Discrete q-Hermite II polynomials, 91, 100,
107
Dual q-Hahn polynomials, 60, 94, 96, 102
Dual q-Krawtchouk polynomials, 79, 94, 96,
99, 104
Dual Hahn polynomials, 31, 44, 45, 102
Gegenbauer polynomials, 34, 103
Hahn polynomials, 29, 43, 45, 102
Hermite polynomials, 41, 45{47, 105{107
Jacobi polynomials, 33, 43{46, 102, 103
Krawtchouk polynomials, 38, 45, 46, 104
Laguerre polynomials, 39, 45{47, 103{105
Legendre polynomials, 37, 102{104
Little q-Jacobi polynomials, 73, 95, 97, 98,
103, 104
Little q-Laguerre polynomials, 84, 97, 99, 105
Little q-Legendre polynomials, 74, 104
Meixner polynomials, 38, 45, 46, 104
Meixner-Pollaczek polynomials, 32, 44, 45,
102, 103
q-Charlier polynomials, 87, 98, 99, 106
q-Hahn polynomials, 59, 93, 95, 96, 102
q-Krawtchouk polynomials, 77, 94, 96, 98,
104
q-Laguerre polynomials, 84, 97{99, 105, 106
q-Meixner polynomials, 75, 95, 98, 104
q-Meixner-Pollaczek polynomials, 64, 95, 96,
103
q-Racah polynomials, 52, 93, 94, 101
q-Scheme, 48, 49
Quantum q-Krawtchouk polynomials, 76, 95,
98, 104
Racah polynomials, 25, 43, 44, 101
Rogers polynomials, 68, 93, 96, 97, 103
Spherical polynomials, 37, 102{104
Stieltjes-Wigert polynomials, 90, 99, 107
Ultraspherical polynomials, 34, 103
Wall polynomials, 84, 97, 99, 105
Wilson polynomials, 23, 43, 101
121
The Askey-scheme of hypergeometric orthogonal polynomials
and its q -analogue
Roelof Koekoek
Rene F. Swarttouw
February 20, 1996
List of errata in report no. 94-05
Page 7, line 1. Replace "all sets" by "all known sets".
Page 7, line -1. Replace "positive denite" by "positive".
Page 12, line -2. This should read
lim
q "1 r s
qa1 ; : : : ; qar q; (q
qb1 ; : : : ; qbs 1)
1+
s rz
= r Fs
a1 ; : : : ; ar z :
b1; : : : ; bs Page 28, formula (1.4.2). The right-hand side should read
(n + a + c) (n + a + d) (n + b + c) (n + b + d)
(2n + a + b + c + d 1) (n + a + b + c + d 1)n! mn :
Page 51, formula (3.1.5). This should read
(1
h
q)2 Dq w~(x; aq 2 ; bq 2 ; cq 2 ; dq 2 jq)Dq y(x)
where
1
1
w~(x; a; b; c; djq) :=
Page 55, formula (3.3.5). This should read
(1
h
1
1
w(x; a; b; c; djq)
p 2 :
1 x
+ 4q n+1(1
1
w~(x; a; b; cjq) :=
(1
h
i
+
qn )w~ (x; a; b; cjq)y(x) = 0; y(x) = pn (x; a; b; cjq);
Page 56, formula (3.4.4). This should read
where
1
q)2 Dq w~ (x; aq 2 ; bq 2 ; cq 2 jq)Dq y(x)
where
i
+
+ nw~(x; a; b; c; djq)y(x) = 0; y(x) = pn(x; a; b; c; djq);
1
w(x; a; b; cjq)
p 2 :
1 x
q)2 Dq w~(x; aq 2 ; bq 2 ; cq 2 ; dq 2 ; q)Dq y(x)
i
+
+ nw~(x; a; b; c; d; q)y(x) = 0; y(x) = pn(x; a; b; c; d; q);
1
1
1
w~(x; a; b; c; d; q) :=
1
1
w(x; a; b; c; d; q)
p 2 :
1 x
Page 57, formula (3.5.2). Replace ( ac) n by ( acq2 )n .
Page 58, line 9. The weight function should read
(c
qx; d 1qx; q)1
(ac 1 qx; bd 1qx; q)1 dq x:
1
Page 58, formula (3.5.8). Replace ( c) n by (
Page 63, formula (3.8.5). This should read
(1
h
q)2 Dq w~(x; aq 2 ; bq 2 jq)Dq y(x)
1
1
+ 4q n+1(1
where
cq2 )n .
i
+
qn)w~(x; a; bjq)y(x) = 0; y(x) = Qn(x; a; bjq);
w~ (x; a; bjq) :=
w(x; a; bjq)
p 2 :
1 x
Page 64, formula (3.9.4). This should read
(1
h
q)2 Dq w~ (x; aq 2 jq)Dq y(x)
1
i
where
+ 4q n+1(1
w~(x; ajq) :=
qn)w~(x; ajq)y(x) = 0; y(x) = Pn (x; ajq);
w(x; ajq)
p 2:
1 x
Page 66, formula (3.10.7). This should read
(1
h
q)2 Dq w~ (x; q+ 2 ; q+ 2 jq)Dq y(x)
1
1
where
i
+ n w~(x; q; q jq)y(x) = 0; y(x) = Pn(;)(xjq);
w~(x; q; q jq) :=
w(x; q; q jq)
p 2 :
1 x
Page 66, fomula (3.10.8). This should read
(1
h
q)2Dq w~(x; q+ 2 ; q+ 2 ; q)Dq y(x)
1
1
where
i
+ n w~ (x; q; q ; q)y(x) = 0; y(x) = Pn(;)(x; q);
w~(x; q; q ; q) :=
w(x; q; q ; q)
p 2 :
1 x
Page 68, formula (3.10.18). This should read
(1
h
q)2 Dq w~(x; q 2 jq)Dq y(x)
1
where
i
+ n w~(x; jq)y(x) = 0; y(x) = Cn (x; jq);
w~(x; jq) :=
w(x; jq)
p 2:
1 x
Page 70, formula (3.10.27). This can be written as
2(1
q2n+1)xPn(x; q) = q
1
2
(1
q2n+2)Pn+1 (x; q) + q 2 (1
1
2
q2n)Pn
1
(x; q):
Page 70, formula (3.10.28). This should read
(1
q)2 Dq [w~(x; q; q)Dq y(x)] + nw~(x; q 2 ; q)y(x) = 0; y(x) = Pn(x; q);
1
where
w(x; a; q)
p 2:
1 x
w~(x; a; q) :=
Page 71, formula (3.11.2). Replace ( ab) n by (
Page 80, formula (3.18.5). This should read
h
i
q)2 Dq w~ (x; aq 2 jq)Dq y(x)
(1
1
where
+ 4q n+1(1
abq2 )n.
qn)w~(x; ajq)y(x) = 0; y(x) = Hn (x; ajq);
w(x; ajq)
p 2:
1 x
w~(x; ajq) :=
Page 82, formula (3.19.5). This can also be written as
2xPn()(xjq) = q 21 14 (1 qn+1)Pn(+1) (xjq) +
+ qn+ 12 + 41 (1 + q 12 )Pn()(xjq) + q 21 + 14 (1
qn+)Pn()1(xjq):
Page 82, formula (3.19.6). This can also be written as
2xPn()(x; q) = q 12 (1 q2n+2)Pn(+1) (x; q) +
+ q2n++ 21 (1 + q)Pn() (x; q) + q 21 (1
q2n+2)Pn()1(x; q):
Page 82, formula (3.19.7). This should read
h
q)2 Dq w~(x; q+ 2 jq)Dq y(x)
(1
1
where
i
+ 4q n+1 (1
w~(x; qjq) :=
Page 82, formula (3.19.8). This should read
(1
h
q)2 Dq w~ (x; q+ 2 ; q)Dq y(x)
1
where
i
w(x; qjq)
p 2:
1 x
+ 4q n+1(1
w~(x; q; q) :=
Page 87, formula (3.24.4). This should read
(1
qn )w~ (x; qjq)y(x) = 0; y(x) = Pn()(xjq);
qn )x2y(x) = aqn 1y(qx)
+ qn (1
x)(a
qn)w~(x; q; q)y(x) = 0; y(x) = Pn()(x; q);
w(x; q; q)
p 2 :
1 x
aqn
+ qn (1 x)(a x) y(x) +
x)y(q 1 x); y(x) = Un(a) (x; q):
1
Page 87, formula (3.25.4). This should read
(1
qn )x2y(x) = (1 x)(a x)y(qx) [(1 x)(a
+ aqy(q 1 x); y(x) = Vn(a) (x; q):
Page 88, formula (3.26.2). Replace d by dx.
3
x) + aq] y(x) +
Page 88, formula (3.26.4). This should read
(1
q)2 Dq [w~(x)Dq y(x)] + 4q n+1(1
where
w~(x) :=
qn)w~(x)y(x) = 0; y(x) = Hn(xjq);
pw(x) 2 :
1
x
Page 90, formula (3.28.4). This should read
q n+1 x2 y(x) = y(qx)
(1 + q)y(x) + q(1
x2 )y(q
1
x); y(x) = hn (x; q):
Page 90, formula (3.29.1). This should read
h~ n (x; q) = i nVn(
1)
(ix; q) =
=
i nq
xn21
(n2 ) 2 0
q n ; ix q n ; q n+1 0
n
q; q
q2
:
x2
2
q ;
Page 90, formula (3.29.2). This can be written as
1 h
X
i
~ m ( cqk ; q)~hn( cqk ; q) w(cqk )qk
h~ m (cqk ; q)~hn(cqk ; q) + h
=
where
k= 1
2
2
2
2
2 ((qq;; cc2;q; c c2q2q;;qq2))1 (qq;nq2)n mn ; c > 0;
1
1
(ix; q)1 ( ix; q)1 = (
Page 90, formula (3.29.4). This should read
w(x) =
(1
qn)x2 y(x) = (1 + x2)y(qx)
1
x2; q2)1
(1 + x2 + q)y(x) + qy(q
:
1
x); y(x) = ~hn(x; q):
Page 109. Reference [40] : "bf A 25" should read A 25.
Acknowledgement
We thank G. Gasper, J. Koekoek, H.T. Koelink, and T.H. Koornwinder for pointing us to
some of these errata.
4