Journal of Difference Equations and Applications,
Vol. 9, No. 9, September 2003, pp. 777–804
On Fourth-order Difference Equations for
Orthogonal Polynomials of a Discrete Variable:
Derivation, Factorization and Solutions
M. FOUPOUAGNIGNIa,*, W. KOEPFa and A. RONVEAUXb
a
Department of Mathematics and Computer Science, University of Kassel, Heinrich-Plett
Straße 40, 34132 Kassel, Germany; bFacultés Universitaires Notre Dame de la Paix,
B-5000 Namur, Belgium
(Received 25 September 2002; Revised 11 January 2003; In final form 23 January 2003)
We derive and factorize the fourth-order difference equations satisfied by orthogonal polynomials obtained from
some modifications of the recurrence coefficients of classical discrete orthogonal polynomials such as: the
associated, the general co-recursive, co-recursive associated, co-dilated and the general co-modified classical
orthogonal polynomials. Moreover, we find four linearly independent solutions of these fourth-order difference
equations, and show how the results obtained for modified classical discrete orthogonal polynomials can be extended
to modified semi-classical discrete orthogonal polynomials. Finally, we extend the validity of the results obtained for
the associated classical discrete orthogonal polynomials with integer order of association from integers to reals.
Keywords: Classical and semi-classical orthogonal polynomials of a discrete variable; D-Laguerre–Hahn class;
Functions of the second kind; Perturbed orthogonal polynomials; Second- and fourth-order difference equations
MSC 2000: Primary 33C45; Secondary 13P05
1. INTRODUCTION
Let U be a regular linear functional [3] on the linear space P of polynomials with real
coefficient and (Pn )n a sequence of monic polynomials, orthogonal with respect to U, i.e.
(i) Pn ðxÞ ¼ x n þ lower degree terms;
(ii) kU; Pn Pm l ¼ kn dn;m ; kn – 0; n [ N;
where N ¼ {0; 1; . . .} denotes the set of non-negative integers. Here, k·,·l means the duality
bracket and dn;m the Kronecker symbol.
(Pn )n satisfies a three-term recurrence equation
Pnþ1 ðxÞ ¼ ðx 2 bn ÞPn ðxÞ 2 gn Pn21 ðxÞ;
n $ 0;
ð1Þ
*Corresponding author. Address: Department of Mathematics, Advanced School of Education, University of
Yaounde I, C/O: P.O. Box 8422 Yaounde, Cameroon. E-mail: [email protected];
[email protected]; [email protected]
ISSN 1023-6198 print/ISSN 1563-5120 online q 2003 Taylor & Francis Ltd
DOI: 10.1080/1023619031000097035
778
M. FOUPOUAGNIGNI et al.
with the initial conditions
P21 ðxÞ ¼ 0;
P0 ðxÞ ¼ 1;
ð2Þ
where bn and gn are real numbers with gn – 0; ;n [ N.0 and N.0 denotes the set
N.0 ¼ {1,2,. . .}.
When the polynomial sequence (Pn)n is classical discrete [28,29], i.e. orthogonal with
respect to a positive weight function r defined on the set I ¼ {a; a þ 1; . . .; b 2 1} and
satisfying the first-order difference equation (called Pearson-type difference equation):
DðsrÞ ¼ tr;
ð3Þ
with
x ¼b
x n s ðxÞrðxÞjx ¼a ¼ 0;
;n [ N;
ð4Þ
each Pn satisfies the difference equation
Ln ðyðxÞÞ ¼ s ðxÞD7yðxÞ þ t ðxÞDyðxÞ þ ln yðxÞ ¼ 0;
where
n
ln ¼ 2 ððn 2 1Þs 00 þ 2t 0 Þ:
2
s is a polynomial of degree at most two and t a first degree polynomial; the operators D and
7 are forward and backward difference operators defined by
DPðxÞ ¼ Pðx þ 1Þ 2 PðxÞ;
7PðxÞ ¼ PðxÞ 2 Pðx 2 1Þ ;P [ P:
The previous difference equation written in terms of forward and backward operators can
be rewritten in terms of the shift operators as
Dn ð yðxÞÞ ¼ ðTLn Þð yðxÞÞ ¼ ððs ðx þ 1Þ þ t ðx þ 1ÞÞT2 2 ð2s ðx þ 1Þ
þ t ðx þ 1Þ 2 ln ÞT þ s ðx þ 1ÞIÞyðxÞ ¼ 0;
n$0
ð5Þ
where T and I are the shift and the identity operators defined, respectively, by
TPðxÞ ¼ Pðx þ 1Þ;
IPðxÞ ¼ PðxÞ ;P [ P:
The orthogonality condition (ii) reads as
b21
X
r ðsÞPn ðsÞPm ðsÞ ¼ kn dn;m ;
kn – 0; ;n [ N:
s¼a
The coefficients bn, gn and ln are given in Refs. [28,29] for any family of classical
discrete orthogonal polynomials and in Refs. [12,16,18,34] in the generic case.
The classical discrete families are Hahn, Kravchuk, Meixner and Charlier orthogonal
polynomials [28].
Some modification of the recurrence coefficients (bn)n and (gn)n of the Eq. (1) lead to new
families of orthogonal polynomials (see Refs. [23,24,33] and references therein) such as the
associated, the general co-recursive, co-recursive associated, co-dilated and the general
co-modified classical discrete orthogonal polynomials [23]. Each of these new families of
orthogonal polynomials satisfy a common fourth-order linear homogeneous difference
FOURTH-ORDER DIFFERENCE EQUATIONS
779
equation with polynomial coefficients of bounded degree. In general, they cannot satisfy a
common second-order linear homogeneous difference equation with polynomial coefficients
of bounded degree. Therefore, these new polynomials are not semi-classical but belong to the
discrete Laguerre – Hahn class (see “Preliminaries and notations” section). Many works have
been devoted to the derivation of these fourth-order difference equations. Their polynomial
coefficients have been given explicitly in Refs. [1,7,8,10,19,32,36] for the rth associated
classical discrete orthogonal polynomials.
In 1999, using symmetry properties inside the three-term recurrence relation,
hypergeometric representation and symbolic computation, the coefficients of the fourthorder difference equation for the co-recursive associated Meixner and Charlier orthogonal
polynomials were given [20].
Despite the fact that the coefficients of the fourth-order difference equation satisfied by the
perturbed classical discrete orthogonal polynomials require heavy computations for being
very large, we have succeeded in deriving and factorizing these fourth-order difference
equations and also finding a basis of four linearly independent solutions of all the difference
equations satisfied by perturbed systems of the classical discrete orthogonal polynomials
considered. Moreover, we have given explicitly the coefficients of the fourth-order difference
equation satisfied by the rth associated classical discrete orthogonal polynomials in terms of
the polynomials s and t appearing in Eq. (3). Also, we have found interesting relations
between the perturbed polynomials, the starting ones and the functions of the second kind (see
the next section for the definition). Therefore, the results obtained in the framework of this
paper are more general and complete the known results in this area. In fact, we deal not only
with the derivation of the fourth-order difference equation for the associated and the corecursive associated classical discrete orthogonal polynomials but with the derivation, the
factorization and the solution basis of the fourth-order difference equations satisfied by the
orthogonal polynomials obtained from some modifications of the recurrence coefficients of
classical discrete orthogonal polynomials as was done for the continuous case [9]. Some
examples of these families are the rth associated, the generalized co-recursive, the generalized
co-dilated, the generalized co-recursive associated and the generalized co-modified classical
discrete orthogonal polynomials.
In the second section, we recall definitions and known results needed for this work. The third
section is devoted to the derivation and the factorization of the fourth-order difference
equation. In the fourth section, we solve difference equations and represent the perturbed
classical orthogonal polynomials in terms of solutions of second-order difference equations.
In the fifth section, we first give hypergeometric representation of solutions of Eq. (5) and
difference operators FðrÞ ; SðrÞ and TðrÞ for the rth associated Charlier and Meixner orthogonal
polynomials; secondly, we extend the results obtained for the associated orthogonal
polynomials with integer order of association from integers to reals. Finally, we show how the
results obtained for modified classical discrete orthogonal polynomials can be extended to
modified semi-classical discrete orthogonal polynomials (see the next section for the
definition).
2. PRELIMINARIES AND NOTATIONS
In this section, we first define the semi-classical and the Laguerre – Hahn class of a given
family of orthogonal polynomials of a discrete variable. Next, we present the families of
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M. FOUPOUAGNIGNI et al.
associated, generalized co-recursive, generalized co-recursive associated, generalized
co-dilated and generalized co-modified orthogonal polynomials, and give relations between
the new sequences and the starting ones.
Each linear functional U generates a so-called Stieltjes function S of U defined by
SðzÞ ¼ 2
X kU; x n l
n$0
z nþ1
;
ð6Þ
where kU; x n l are the moments of the functional U. The linear functional U satisfies in
general a simple functional equation living in P0 , the dual space of P. Appropriate
definitions of DðUÞ and PU, where P is a polynomial that allows building a simple difference
equation for the functional, which generalizes in some way the Pearson-type difference
equation for the weight r [7,10,13,34].
If the Stieltjes function S(x) satisfies a first-order linear difference equation of the form
fðxÞ Sðx þ 1Þ ¼ CðxÞ SðxÞ þ DðxÞ;
ð7Þ
where f, C and D are polynomials, the functional U satisfies in P0 a first-order difference
equation with polynomial coefficients. In this case, the functional U and the corresponding
orthogonal polynomial sequence (Pn)n belong to the discrete semi-classical class (and are
therefore called semi-classical discrete) which includes the classical discrete families
[7,10,13,14,25,26,34].
Each semi-classical discrete orthogonal polynomials sequence (Pn)n satisfies a common
second-order difference equation [7,10,13,14,22,25,26,34]
Mn ð yðxÞÞ ¼ ðI 2 ðx; nÞT2 þ I 1 ðx; nÞT þ I 0 ðx; nÞIÞ yðxÞ ¼ 0;
ð8Þ
where Ii(x, n) are polynomials in x of degree not depending on n. Notice that this secondorder difference equation for the semi-classical discrete orthogonal polynomials appears in
Ref. [10] as D0 ;n ðyÞ ¼ 0 (using Equations 3.16 and 3.20).
An important class, larger than the semi-classical discrete one, appears when the Stieltjes
function satisfies a D-Riccati difference equation [7,10,13]
fðx þ 1Þ DSðxÞ ¼ GðxÞ SðxÞ Sðx þ 1Þ þ EðxÞSðxÞ þ FðxÞ Sðx þ 1Þ þ HðxÞ;
ð9Þ
where f – 0; G, E, F and H are polynomials fulfilling a certain conditions (see Ref. [11],
Eq. 15). The corresponding functional U satisfies then a complicated quadratic difference
equation in P0 : U and the corresponding orthogonal polynomial families are said to belong to
the discrete Laguerre – Hahn class [7,10,13], denoted as D-Laguerre– Hahn class.
It is well known that any D-Laguerre – Hahn orthogonal polynomial sequence satisfies a
common fourth-order difference equation of the form [7,10]
ðJ 4 ðx; nÞT4 þ J 3 ðx; nÞT3 þ J 2 ðx; nÞT2 þ J 1 ðx; nÞT þ J 0 ðx; nÞIÞyðxÞ ¼ 0;
where Ji(x,n) are polynomials of degree not depending on n.
Furthermore, it is known that many perturbations of the recurrence coefficients of any
Laguerre –Hahn family generate orthogonal polynomials belonging to the Laguerre – Hahn
class and, therefore, satisfy a fourth-order differential or difference equation [7,10,13,21,34].
FOURTH-ORDER DIFFERENCE EQUATIONS
781
2.1. Perturbation of Recurrence Coefficients
Now we consider a sequence of polynomials (Pn)n, orthogonal with respect to a regular linear
functional U, satisfying Eq. (1). Orthogonal families we will deal with are the associated
orthogonal polynomials and those obtained from finite modification of the recurrence
coefficients in Eq. (1). Some examples of these families are:
Þ
2.1.1. The Associated Orthogonal Polynomials ðP ðr
n Þn
Given r [ N; the rth associated of the polynomials (Pn)n, is a polynomial sequence denoted
by ðPðrÞ
n Þn and defined by the recurrence equation (1) in which bn and gn are replaced by bnþr
and gnþr, respectively
ðrÞ
ðrÞ
PðrÞ
nþ1 ðxÞ ¼ ðx 2 bnþr Þ Pn ðxÞ 2 gnþr Pn21 ðxÞ;
n$1
ð10Þ
with the initial conditions
PðrÞ
21 ðxÞ ¼ 0;
PðrÞ
0 ðxÞ ¼ 1:
ð11Þ
The family ðPðrÞ
n Þn ; thanks to Favard’s theorem [6] (see also Ref. [3]), is orthogonal.
It is related to the starting polynomials and its first associated by the relation [4]
PðrÞ
n ðxÞ ¼
Pr21 ðxÞ ð1Þ
Pð1Þ ðxÞ
Pnþr21 ðxÞ 2 r22 Pnþr ðxÞ;
Gr21
Gr21
n $ 0;
r $ 1;
ð12Þ
where the sequence (Gn)n is defined by
Gn ¼
n
Y
gi ;
n $ 1;
ð13Þ
G0 ; 1:
i¼1
2.1.2. The Co-recursive ðP ½nm Þn and the Generalized Co-recursive Orthogonal
;m
Polynomials ðP ½k
Þn
n
The co-recursive of the orthogonal polynomial (Pn)n, denoted by ðP½nm Þn ; was introduced for
the first time by Chihara [2], as the family of polynomials generated by the recursion formula
(1) in which b0 is replaced by b0 þ m :
m
m
P½nþ1
ðxÞ ¼ ðx 2 bn ÞP½nm ðxÞ 2 gn P½n21
ðxÞ;
n $ 1;
ð14Þ
with the initial conditions
P½0m ðxÞ ¼ 1;
P1½m ðxÞ ¼ x 2 b0 2 m;
ð15Þ
where m denotes a real number.
This notion was extended to the generalized co-recursive orthogonal polynomials in
Refs. [4,5,31] by modifying the sequence (bn)n at the level k. This yields an orthogonal
polynomial sequence denoted by ðPn½k;m Þn and generated by the recursion formula
m
* ½k;m ðxÞ 2 g P½k;m ðxÞ;
P½k;
n n21
nþ1 ðxÞ ¼ ðx 2 bn ÞPn
n $ 1;
ð16Þ
782
M. FOUPOUAGNIGNI et al.
with the initial conditions
P0½k;m ðxÞ ¼ 1;
P1½k;m ðxÞ ¼ x 2 b*0 ;
ð17Þ
*
where b*
n ¼ bn for n – k and bk ¼ bk þ m:
The orthogonal polynomial sequence ðPn½k;m Þn is related to (Pn)n and is associated by
Ref. [23]
ðkþ1Þ
m
P½k;
ðxÞ ¼ Pn ðxÞ 2 mPk ðxÞPn2ðkþ1Þ
ðxÞ;
n
m
ðxÞ ¼ Pn ðxÞ;
P½k;
n
n $ k þ 1;
ð18Þ
n # k:
Use of Eq. (12) transforms the previous equations in
m
ðxÞ
P½k;
n
!
mP2k ðxÞ ð1Þ
mPk ðxÞPð1Þ
k21
¼2
Pn21 ðxÞ þ 1 þ
Pn ðxÞ;
Gk
Gk
m
ðxÞ ¼ Pn ðxÞ;
P½k;
n
n $ k þ 1;
ð19Þ
n # k:
m
Obviously, we have the relations P½0;
¼ P½nm and P½0
n
n ðxÞ ¼ Pn :
;m}
Þn and the Generalized Co-recursive
2.1.3. The Co-recursive Associated ðP {r
n
{r ;k ;m}
Associated Orthogonal Polynomials ðP n
Þn
The co-recursive associated as well as the generalized co-recursive associated of the
m}
orthogonal polynomial sequence (Pn)n, denoted by ðP{r;
Þn and ðPn{r;k;m} Þn ; respectively, are,
n
the co-recursive and the generalized co-recursive (with modification on bk) of the associated
ðPðrÞ
n Þn of (Pn)n, respectively. Thanks to Eq. (18), they are related with (Pn)n and is associated by
m}
m}
¼ P{r;
;
P{r;0;
n
n
and
ðrÞ
ðrþkþ1Þ
m}
ðxÞ ¼ PðrÞ
P{r;k;
n
n ðxÞ 2 mPk ðxÞPn2ðkþ1Þ ðxÞ;
m}
ðxÞ ¼ PðrÞ
P{r;k;
n
n ðxÞ;
n $ k þ 1;
ð20Þ
n # k:
The generalized co-recursive associated orthogonal polynomials can also be expressed
using Eqs. (12) and (20) by
!
Pr21 ðxÞ mPkþr ðxÞPðrÞ
ðxÞ
k
2
Pð1Þ
nþr21 ðxÞ
Gr21
Grþk
m}
ðxÞ ¼
P{r;k;
n
2
!
ð1Þ
ðrÞ
m
P
ðxÞP
ðxÞ
Pð1Þ
ðxÞ
kþr21
k
r22
2
Pnþr ðxÞ;
Gr21
Grþk
m}
ðxÞ ¼ PðrÞ
P{r;k;
n
n ðxÞ;
n # k:
n $ k þ 1;
ð21Þ
FOURTH-ORDER DIFFERENCE EQUATIONS
783
2.1.4. The Co-dilated ðP jlj
n Þn and the Generalized Co-dilated Orthogonal
;lj
Polynomials ðP jk
Þ
n
n
The co-dilated of the orthogonal polynomial sequence (Pn)n, denoted by ðPnjlj Þn ; was
introduced by Dini [4], as the family of polynomials generated by the recursion formula (1)
in which g1, is replaced by l g1, i.e.
j lj
j lj
ðxÞ ¼ ðx 2 bn ÞPnjlj ðxÞ 2 gn Pn21
ðxÞ;
Pnþ1
n$2
ð22Þ
with the initial conditions
P0jlj ðxÞ ¼ 1;
P1jlj ðxÞ ¼ x 2 b0 ; P2jlj ðxÞ ¼ ðx 2 b0 Þðx 2 b1 Þ 2 lg1 ;
ð23Þ
where l is a non-zero real number.
This notion was extended to the generalized co-dilated polynomials in Refs. [5,31] by
modifying the sequence (gn)n at the level k. This yields an orthogonal polynomial sequence
lj
denoted by ðPjk;
n Þn and generated by the recurrence equation
lj
jk;lj
* jk;lj
Pjk;
nþ1 ðxÞ ¼ ðx 2 bn ÞPn ðxÞ 2 gn Pn21 ðxÞ;
n $ 1;
ð24Þ
with the initial conditions
lj
Pjk;
0 ðxÞ ¼ 1;
lj
Pjk;
1 ðxÞ ¼ x 2 b0 ;
ð25Þ
*
where g*
n ¼ gn for n – k and gk ¼ lgk :
lj
The orthogonal polynomial sequence ðPjk;
n Þn is related to (Pn)n and is associated by
Ref. [23]
ðkþ1Þ
lj
Pjk;
n ðxÞ ¼ Pn ðxÞ þ ð1 2 lÞgk Pk21 ðxÞPn2ðkþ1Þ ðxÞ;
n $ k þ 1;
ð26Þ
lj
Pjk;
n ðxÞ ¼ Pn ðxÞ;
n # k:
Use of Eq. (12) transforms the previous equation in
!
ð1 2 lÞPk21 ðxÞPð1Þ
ð1 2 lÞPk21 ðxÞPk ðxÞ ð1Þ
jk;lj
k21
Pn ðxÞ ¼ 1 2
Pn21 ðxÞ; n $ k þ 1;
Pn ðxÞ þ
Gk21
Gk21
lj
Pjk;
n ðxÞ ¼ Pn ðxÞ;
ð27Þ
n # k:
For k ¼ 1 or l ¼ 1; we have
Pnj1;lj ¼ Pnjlj ;
Pjk;1j
¼ Pn :
n
;m;l
2.1.5. The Generalized Co-modified Orthogonal Polynomials ðP ½k
Þn
n
New families of orthogonal polynomials can also be generated by modifying at the same time
the sequences (bn)n and (gn)n at the levels k and k0 , respectively. When k ¼ k0 ; the new family
obtained [23], denoted by ðPn½k;m;l Þn is generated by the three-term recurrence relation
m; l * ½k;m;l ðxÞ 2 g* P½k;m;l ðxÞ;
P½k;
n n21
nþ1 ðxÞ ¼ ðx 2 bn ÞPn
n $ 1;
ð28Þ
784
M. FOUPOUAGNIGNI et al.
with the initial conditions
P0½k;m;l ðxÞ ¼ 1;
P1½k;m;l ðxÞ ¼ x 2 b*0 ;
ð29Þ
*
*
*
where b*
n ¼ bn ; gn ¼ gn for n – k and bk ¼ bk þ m; gk ¼ lgk : This family is represented
in terms of the starting polynomials and their associated by Ref. [23]
m; l P½k;
ðxÞ ¼ Pn ðxÞ þ ðð1 2 lÞgk Pk21 ðxÞ 2 mPk ðxÞÞPðkþ1Þ
n
n2ðkþ1Þ ðxÞ;
lj
Pjk;
n ðxÞ ¼ Pn ðxÞ;
n $ k þ 1;
ð30Þ
n # k:
The latter relation can also be written as
Pn½k;m;l ðxÞ
!
ð1 2 lÞPk21 ðxÞPð1Þ
mPk ðxÞPð1Þ
k21
k21 ðxÞ
12
þ
Pn ðxÞ
Gk
Gk21
¼
ð1 2 lÞPk21 ðxÞPk ðxÞ mP2k ðxÞ ð1Þ
þ
2
Pn21 ðxÞ;
Gk21
Gk
Pn½k;l ðxÞ ¼ Pn ðxÞ;
n $ k þ 1;
ð31Þ
n # k:
2.2. Results on Classical Discrete Orthogonal Polynomials
Next, we state the following lemmas which are essential for this work. The first one is due to
Atakishiyev, Ronveaux and Wolf [1] but the representation with the shift operator given by
Eq. (32) is taken from Ref. [10] (see also Ref. [32]).
Lemma 1 [1] Given a classical discrete orthogonal polynomial sequence (Pn)n satisfying
Eq. (5), the following relation holds
00
ð1Þ
s
*
0
Dn Pn21 ðxÞ ¼
2 t ðð2sð1Þ þ tð1Þ 2 ln ÞT 2 ð2sð1Þ þ tð1Þ ÞIÞPn ðxÞ;
ð32Þ
2
where the operator D* is given by
n
Dn* ¼ ðsð1Þ þ tð1Þ Þ ðsð2Þ T2 2 ð2sð1Þ þ tð1Þ 2 ln ÞT þ ðs þ tÞIÞ
ð33Þ
and
s ; s ðxÞ;
t ; t ðxÞ;
sð1Þ ; s ðx þ 1Þ;
tð1Þ ; t ðx þ 1Þ;
sð2Þ ; s ðx þ 2Þ:
ð34Þ
It should be noticed that Dn and D*n are related by
sð1Þ D*n ðryÞ ¼ r ðs þ tÞðsð1Þ þ tð1Þ Þ Dn ð yÞ;
;y;
where r is the weight function satisfying Eqs. (3) and (4).
Lemma 2
[28]
1. Two linearly independent solutions of the difference equation
Ln ðyðxÞÞ ¼ s ðxÞD7 yðxÞ þ tðxÞDyðxÞ þ ln yðxÞ ¼ 0;
ð35Þ
FOURTH-ORDER DIFFERENCE EQUATIONS
785
are Pn and Qn, where (Pn)n is a polynomial sequence, orthogonal with respect to the
weight function r defined on the set I ¼ {a; a þ 1; . . .; b 2 1}; satisfying Eqs. (3) and (4).
The constants ln is given by
n
ln ¼ 2 ððn 2 1Þs 00 þ 2t 0 Þ;
2
while Qn is the function of the second kind, defined by
Qn ðxÞ ¼
b21
1 X
rðsÞPn ðsÞ
;
rðxÞ s¼a s 2 x
x {a; a þ 1; . . .; b 2 1}:
ð36Þ
When x ¼ t [ {a; a þ 1; . . .; b 2 1}; then Qn(t) is defined by
Qn ðtÞ ¼
b21
X
1
rðsÞPn ðsÞ
:
rðtÞ a#s#b21; s–t s 2 t
ð37Þ
2. The polynomials Pn and the function Qn are two linearly independent solutions of
the recurrence equation (1).
3. FACTORIZATION OF FOURTH-ORDER DIFFERENCE OPERATORS
Given (Pn)n a classical discrete orthogonal polynomial sequence, we consider in general
all transformations which lead to new families of orthogonal polynomials denoted by ðP n Þn
and are related to the starting sequence by
P n ðxÞ ¼ An ðxÞPð1Þ
nþk21 þ Bn ðxÞPnþk ;
n $ k0 ;
ð38Þ
where An and Bn are polynomials of degree not depending on n, and k, k0 [ N: (Among
these transformations are the associated orthogonal polynomials and those obtained from
finite modification of the recurrence coefficients of Eq. (1). Some examples are listed in
Subsection 2.1).
We have the following:
Theorem 1
1. The orthogonal polynomials ðP n Þn$k0 satisfy a common fourth-order linear difference
equation
Fn ðyðxÞÞ ¼ ðJ4 ðx;nÞT4 þ J3 ðx;nÞT3 þ J2 ðx;nÞT2 þ J1 ðx;nÞT þ J0 ðx;nÞIÞyðxÞ ¼ 0; ð39Þ
where the coefficients Ji are polynomials in x, with degree not depending on n.
2. The operator Fn can be factored as product of two-second order linear difference
operators Sn and Tn :
Xn Fn ¼ Sn Tn ; n $ k
ð40Þ
where Xn is a polynomial of fixed degree, depending on Pr21, s and t, and the coefficients
in Sn and Tn are polynomials of degree not depending on n.
786
M. FOUPOUAGNIGNI et al.
Proof
In the first step, we solve Eq. (38) in terms of Pð1Þ
nþk21
Pð1Þ
nþk21 ðxÞ ¼
P n ðxÞ 2 Bn ðxÞPnþk ðxÞ
An ðxÞ
ð41Þ
and substitute the previous relation in Eq. (32) in which n is replaced by n þ k. Then we use
Eq. (5) (for Pnþk) to eliminate the term T2 Pnþk and get
Mnþk ðP n Þ ¼ b1 TPnþk þ b0 Pnþk ;
ð42Þ
where bi are rational functions and Mnþk a second-order linear difference operator given in
* (see Eq. (32)) by
terms of operator Dnþk
y
Mnþk ð yÞ ¼ An ðTAn ÞðT2 An ÞD*nþk
:
ð43Þ
An
Next, we shift Eq. (42) and use again Eq. (5) to eliminate T2 Pnþk ; and get
TMnþk ðP n Þ ¼ c1 TPnþk þ c0 Pnþk :
ð44Þ
We reiterate the same process using the previous equation and get
T2 Mnþk ðP n Þ ¼ d1 TPnþk þ d0 Pnþk ;
ð45Þ
where ci and di are again rational functions.
The fourth-order difference equation is given in determinantal form from Eqs. (42), (44)
and (45)
b1 b0
Mnþk ðP n Þ ð46Þ
Fn ðP n Þ ¼ c1 c0 TMnþk ðP n Þ ¼ 0:
2
d1 d0 T Mnþk ðP n Þ The previous equation can be written as
Fn ðP n Þ ¼ e2 T2 Mnþk ðP n Þ þ e1 TMnþk ðP n Þ þ e0 Mnþk ðP n Þ ¼ ½Sn Tn ðP n Þ ¼ 0;
ð47Þ
where the second-order difference operators Sn and Tn are given by
Sn ¼ e2 T2 þ e1 T þ e0 I;
Tn ¼ Mnþk :
ð48Þ
We conclude the proof by noticing that after cancellation of the denominator in Eq. (46), the
coefficients ei are polynomials of degree not depending on n.
A
We would like to mention that the factorization pointed out in the previous theorem
(except the case of the first associated classical discrete orthogonal polynomials already
treated in Ref. [1] (see, also Ref. [10], equation 4.16 for more details) seems to be a new
results and has lots of applications as will be shown later.
½k;m
lj
m; l In what follows, we will denote, respectively, by FðrÞ
; Fn{r;k;m} ; Fjk;
and F½k;
the
n ; Fn
n
n
fourth-order difference operators for the rth associated, the generalized co-recursive, the
generalized co-recursive associated, the generalized co-dilated, and the generalized
co-modified orthogonal polynomials.
FOURTH-ORDER DIFFERENCE EQUATIONS
787
3.1. Some Consequences
For the rth associated classical discrete orthogonal polynomials ðPðrÞ
n Þn ; we have used the
previous theorem and the representation given in Eq. (12) to compute the operators Sn and
Tn using Maple 8 [27].
Proposition 1 The two difference operator factors of the fourth-order difference operator
for the rth associated classical discrete orthogonal polynomials are
2
SðrÞ
n ¼ sð2Þ Pr21 ðx þ 1Þðtð1Þ þ sð1Þ Þ ð2tð1Þ 2 t þ sð2Þ Þ ð23s2 þ 3s1 2 sÞ
£ ð22tð1Þ þ 8sð1Þ þ z 2 6sð2Þ 2 2s þ 2tÞT2 þ ð2sð2Þ ð8sð1Þ 2 6sð2Þ 2 3sÞ
£ ðtð1Þ þ sð1Þ Þsð1Þ zð2z 2 lr21 2 8sð1Þ 2 3t þ 2s þ 8sð2Þ þ 4tð1Þ ÞPr21 ðxÞ 2 sð2Þ
£ ð8sð1Þ 2 6sð2Þ 2 3sÞðtð1Þ þ sð1Þ Þð6s2ð2Þ sð1Þ þ 2sð2Þ t2ð1Þ 2 6s2 sð1Þ tð1Þ 2 8sð2Þ s12
2
þ 6sð2Þ
tð1Þ þ 2sð2Þ stð1Þ 2 2sð2Þ ttð1Þ 2 2sð2Þ tsð1Þ þ 2sð2Þ sð1Þ s þ 3tztð1Þ þ 6tzsð1Þ
2 3tzlr21 2 2sztð1Þ 2 4szsð1Þ þ 2szlr21 þ 5tð1Þ lr21 z 2 6lr21 sð1Þ z 2 zl2r21
2
2
þ 16zsð1Þ
2 4ztð1Þ
þ z2 tð1Þ þ 2z2 sð1Þ 2 z2 lr21 þ 8lr21 sð2Þ z 2 16sð2Þ sð1Þ z
2 8sð2Þ tð1Þ zÞ Pr21 ðx þ 1ÞÞT 2 sð1Þ ð8sð1Þ 2 6sð2Þ 2 3sÞ ð23sð2Þ þ 3sð1Þ 2 sÞ
£ ð2Pr21 ðx þ 1Þt21 2 3Pr21 ðx þ 1Þlr21 tð1Þ þ t1 Pr21 ðx þ 1Þsð2Þ 2 tð1Þ Pr21 ðx þ 1Þt
þ 4tð1Þ sð1Þ Pr21 ðx þ 1Þ 2 2tð1Þ sð1Þ Pr21 ðxÞ þ 3sð1Þ Pr21 ðx þ 1Þsð2Þ
2 2Pr21 ðx þ 1Þsð2Þ lr21 2 2Pr21 ðx þ 1Þsð1Þ t 2 2sð1Þ lr21 Pr21 ðx þ 1Þ
þ Pr21 ðx þ 1Þlr21 t þ l2r21 Pr21 ðx þ 1Þ 2 2s1 sð2Þ Pr21 ðxÞ þ sð1Þ lr21 Pr21 ðxÞ
þ sð1Þ Pr21 ðxÞ þ sð1Þ Pr21 ðxÞtÞz I;
ð49Þ
2
2
TðrÞ
n ¼ sð2Þ Pr21 ðx þ 1ÞPr21 ðxÞðtð1Þ þ sð1Þ ÞT 2 sð2Þ Pr21 ðxÞð2tð1Þ Pr21 ðx þ 1Þ
2 2sð1Þ Pr21 ðx þ 1Þ þ sð1Þ Pr 2 1ðxÞ þ lr21 Pr21 ðx þ 1ÞÞ ð22sð1Þ þ lnþr 2 tð1Þ ÞÞT
2 sð2Þ Pr21 ðx þ 1Þðs þ tÞð2tð1Þ Pr21 ðx þ 1Þ 2 2sð1ÞPr21 ðx þ 1Þ
þ sð1Þ Pr21 ðxÞ þ lr21 Pr21 ðx þ 1ÞÞI;
ð50Þ
where (Pn)n is the sequence of classical orthogonal satisfying Eq. (5), r [ N.0 and
z ¼ 5tð1Þ 2 6sð1Þ þ 8sð2Þ þ 2s 2 3t 2 lr21 2 lnþr :
Moreover, we have
ðrÞ
ðrÞ
SðrÞ
n Tn ¼ X n ðs; t; Pr21 ; lr21 ÞFn ;
ð51Þ
788
M. FOUPOUAGNIGNI et al.
where
X n ; X n ðs; t; Pr21 ; lr21 Þ ¼ sð2Þ Pr21 ðx þ 1Þ ð23s2 þ 3sð1Þ 2 sÞðtð1Þ Pr21 ðx þ 1Þ
þ 2sð1Þ Pr21 ðx þ 1Þ 2 sð1Þ Pr21 ðxÞ 2 lr21 Pr21 ðx þ 1ÞÞ ð8sð1Þ 2 6sð2Þ 2 3sÞ
£ ð2Pr21 ðx þ 1Þt2ð1Þ 2 3Pr21 ðx þ 1Þlr21 t1 þ tð1Þ Pr21 ðx þ 1Þsð2Þ 2 tð1Þ Pr21 ðx þ 1Þt
þ 4tð1Þ sð1Þ Pr21 ðx þ 1Þ 2 2tð1Þ sð1Þ Pr21 ðxÞ þ 3sð1Þ Pr21 ðx þ 1Þsð2Þ 2 2Pr21 ðx þ 1Þsð2Þ lr21
2 2Pr21 ðx þ 1Þsð1Þ t 2 2sð1Þ lr21 Pr21 ðx þ 1Þ þ Pr21 ðx þ 1Þlr21 t þ l2r21 Pr21 ðx þ 1Þ
2 2sð1Þ sð2Þ Pr21 ðxÞ þ sð1Þ lr21 Pr21 ðxÞ þ sð1Þ Pr21 ðxÞtÞ
and
4
3
2
FðrÞ
n ¼ I 4 ðx; nÞT þ I 3 ðx; nÞT þ I 2 ðx; nÞT þ I 1 ðx; nÞT þ I 0 ðx; nÞI;
ð52Þ
with
I 0 ðx; n; rÞ ¼ 2sð1Þ ðs þ tÞz;
2
2
2
2
I 1 ðx; n; rÞ ¼ 6sð2Þ
s1 þ 2sð2Þ tð1Þ
2 6sð2Þ sð1Þ tð1Þ 2 8sð2Þ sð1Þ
þ 6sð2Þ
tð1Þ þ 2sð2Þ stð1Þ
2 2sð2Þ ttð1Þ 2 2sð2Þ tsð1Þ þ 2s2 sð1Þ s þ 3tztð1Þ þ 6tzsð1Þ 2 3tzlr21 2 2sztð1Þ
2
2 4szsð1Þ þ 2szlr21 þ 5tð1Þ lr21 z 2 6lr21 s1 z 2 zl2r21 þ 16zs12 2 4ztð1Þ
þ z 2 tð1Þ
þ 2z 2 sð1Þ 2 z 2 lr21 þ 8lr21 sð2Þ z 2 16sð2Þ sð1Þ z 2 8sð2Þ t1 z;
I 2 ðx; n; rÞ ¼ 2z 3 þ ð25t þ 14sð2Þ 2 2lr21 þ 4s þ 7t1 2 14sð1Þ Þz 2 þ ð257s2ð2Þ 2 4s 2
þ 12tð1Þ lr21 2 20lr21 sð1Þ 2 8t 2 þ 11st þ 24ttð1Þ þ 41sð2Þ t 2 37sð1Þ t 2 8lr21 t
2
2
2 16stð1Þ 2 31sð2Þ s þ 28sð1Þ s þ 6lr21 s 2 2l2r21 2 48sð1Þ
2 18tð1Þ
þ 54sð1Þ tð1Þ
2 59sð2Þ tð1Þ þ 22lr21 sð2Þ þ 106s2 sð1Þ Þz þ 2ð24sð1Þ 2 t þ s þ 3sð2Þ
þ tð1Þ Þ ð3sð1Þ t þ 6lr21 sð1Þ 2 9sð2Þ sð1Þ 2 6sð1Þ tð1Þ þ 2t 2 þ 3lr21 t þ 3sð2Þ s
2 7sð2Þ t 2 st þ 2stð1Þ 2 7ttð1Þ 2 2lr21 s 2 8lr21 sð2Þ þ 12sð2Þ tð1Þ 2 5tð1Þ lr21
þ 9s2ð2Þ þ l2r21 þ 6t21 Þ;
I 3 ðx; n; rÞ ¼ ð22t þ 6sð2Þ 2 lr21 þ 2s þ 3tð1Þ 2 6s1 Þz 2 2 ð2lr21 2 8sð1Þ 2 3t þ 2s þ 8sð2Þ
þ 4tð1Þ Þ ð22t þ 6s2 2 lr21 þ 2s þ 3tð1Þ 2 6sð1Þ Þz þ 2ð24sð1Þ 2 t þ s þ 3sð2Þ
þ tð1Þ Þ ð3sð1Þ t þ 6lr21 sð1Þ 2 9sð2Þ sð1Þ 2 6sð1Þ tð1Þ þ 2t 2 þ 3lr21 t þ 3sð2Þ s
2 7sð2Þ t 2 st þ 2st1 2 7ttð1Þ 2 2lr21 s 2 8lr21 sð2Þ þ 12sð2Þ tð1Þ 2 5tð1Þ lr21
þ 9s2ð2Þ þ l2r21 þ 6t2ð1Þ Þ;
FOURTH-ORDER DIFFERENCE EQUATIONS
789
I 4 ðx; n; rÞ ¼ 2ð8sð1Þ 2 6sð2Þ 2 3sÞ ð3sð2Þ þ s 2 3sð1Þ þ 3tð1Þ 2 2tÞ ð28sð1Þ þ 2s 2 2t þ 2tð1Þ
þ 6sð2Þ 2 zÞ:
Corollary 1
The fourth-order difference operator can also be factorized as
~ ðrÞ T
~ ðrÞ ¼ Xðs; t; Qr21 ; lr21 ÞFðrÞ ;
S
n
n
n
ð53Þ
~ ðrÞ and T
~ ðrÞ are obtained from
where the expression Xðs; t; Qr21 ; lr21 Þ and the operators S
n
n
ðrÞ
the expression Xðs; t; Pr21 ; lr21 Þ and operators SðrÞ
and
T
;
respectively
by replacing the
n
n
polynomials Pr21 with the function Qr21.
The proof is obtained by a direct computation using that Pn and Qn satisfies Eq. (5).
Proposition 2 The operator Tn for the generalized co-recursive and co-dilated classical
lj
discrete orthogonal polynomials ðPn½k;m Þn and ðPjk;
n Þn (with k $ 1), denoted, respectively by
m
lj
T½k;
; Tjk;
are obtained in the same way:
n
n
m
T½k;
¼ sð2Þ P2k ðxÞP2k ðx þ 1Þðtð1Þ þ sð1Þ Þ2 T2 2 P2k ðxÞðtð1Þ 2 ln þ 2s1 Þ ð2sð1Þ Pk ðxÞ
n
þ 2sð1Þ Pk ðx þ 1Þ 2 lk Pk ðx þ 1Þ þ tð1Þ Pk ðx þ 1ÞÞ2 T þ P2k ðx þ 1Þðs þ tÞ
ð2tð1Þ Pk ðx þ 1Þ 2 2sð1Þ Pk ðx þ 1Þ þ sð1Þ Pk ðxÞ þ lk Pk ðx þ 1ÞÞ2 I;
ð54Þ
lj
Tjk;
¼sð2Þ Pk21 ðx þ 1ÞPk ðx þ 1ÞPk21 ðxÞPk ðxÞðtð1Þ þ sð1Þ Þ2 T2 2 Pk21 ðxÞPk ðxÞ
n
£ ðtð1Þ 2 ln þ 2s1 Þ £ ð2tð1Þ Pk ðx þ 1Þ 2 2sð1Þ Pk ðx þ 1Þ þ s1 Pk ðxÞ þ lk Pk ðx þ 1ÞÞ
£ ð22sð1Þ Pk21 ðx þ 1Þ þ sð1Þ Pk21 ðxÞ 2 Pk21 ðx þ 1Þtð1Þ þ lk21 Pk21 ðx þ 1ÞÞT
þ Pk21 ðx þ 1ÞPk ðx þ 1Þðs þ tÞð2tð1Þ Pk ðx þ 1Þ 2 2sð1Þ Pk ðx þ 1Þ þ sð1Þ Pk ðxÞ
þ lk Pk ðx þ 1ÞÞð22sð1Þ Pk21 ðx þ 1Þ þ sð1Þ Pk21 ðxÞ 2 Pk21 ðx þ 1Þtð1Þ
þ lk21 Pk21 ðx þ 1ÞÞI:
ð55Þ
The operators Sn for the generalized co-recursive and co-dilated classical orthogonal
polynomials are very large expressions; however, they can be obtained using the previous
theorem and Eqs. (21) and (31). The same remark applies for the factors Sn and Tn of the
fourth-order difference equation satisfied by the generalized co-recursive associated and
generalized co-modified classical orthogonal polynomials.
4. SOLUTIONS OF THE FOURTH-ORDER DIFFERENCE EQUATIONS
In the following, we solve the fourth-order difference equation satisfied by the five
perturbations listed in the second section and represent the new families of orthogonal
polynomials in terms of solutions of second-order difference equations.
Theorem 2 Let (Pn)n be a classical discrete orthogonal polynomial sequence, r [ N.0
and ðPðrÞ
n Þn the rth associated of (Pn)n. Four linearly independent solutions of the difference
790
M. FOUPOUAGNIGNI et al.
equation
FðrÞ
n ðyÞ ¼ 0
ð56Þ
ðrÞ
satisfied by ðPðrÞ
n Þn ; where Fn is given by Eq. (52), are
AðrÞ
n ðxÞ ¼ rðxÞPr21 ðxÞPnþr ðxÞ;
BðrÞ
n ðxÞ ¼ rðxÞPr21 ðxÞQnþr ðxÞ;
ð57Þ
CðrÞ
n ðxÞ ¼ rðxÞQr21 ðxÞ Pnþr ðxÞ;
DðrÞ
n ðxÞ ¼ rðxÞQr21 ðxÞQnþr ðxÞ;
Qn denoting the function of second kind associated to (Pn)n which is defined by Eqs. (36)
and (37).
Moreover, PðrÞ
n is related to these solutions by
PðrÞ
n ðxÞ ¼
ðrÞ
BðrÞ
rðxÞðPr21 ðxÞQnþr ðxÞ 2 Qr21 ðxÞPnþr ðxÞÞ
n ðxÞ 2 C n ðxÞ
¼
;
g0 Gr21
g0 Gr21
;n [ N;
ð58Þ
;r [ N.0 ;
where Gk is given by Eq. (13) and g0 defined as
g0 ¼
b21
X
rðsÞ:
ð59Þ
s¼a
Proof
In the first step, we solve the difference equation
TðrÞ
n ðyÞ ¼ 0:
To do this, we use Eqs. (12), (35), (38), (43) and (48) to get
TðrÞ
n ðyÞ ¼ Mnþr ðyÞ
* y ¼ Pr21 ðxÞPr21 ðx þ 1ÞPr21 ðx þ 2ÞDnþr
Pr21
ð60Þ
¼ Pr21 ðxÞPr21 ðx þ 1ÞPr21 ðx þ 2ÞrðxÞðs ðxÞ þ t ðxÞÞ ðsð1Þ þ tð1Þ ÞDnþr ðzÞ=sð1Þ ;
where the functions y and z are related by y ¼ zrPr21 : Since the two linearly independent
solutions of Dnþr ðzÞ ¼ 0 are Pnþr and Qnþr (see Lemma 2), the two linearly independent
solutions of TðrÞ
n ðyÞ ¼ 0 (which are also solutions of Eq. (56) thanks to Eq. (51)) are
AðrÞ
n ðxÞ ¼ rðxÞPr21 ðxÞPnþr ðxÞ;
BðrÞ
n ðxÞ ¼ rðxÞPr21 ðxÞQnþr ðxÞ:
ð61Þ
Use of Eqs. (50) and (53) taking care that the weight function r and the function Qn satisfy
Eqs. (3) and (5), respectively, leads to
~ ðrÞ ðyÞ ¼ Qr21 ðxÞQr21 ðx þ 1ÞQr21 ðx þ 2Þr ðxÞ ðs ðxÞ þ tðxÞÞðsð1Þ þ tð1Þ ÞDnþr ðzÞ=sð1Þ ; ð62Þ
T
n
where the functions y and z are related by y ¼ zrQr21 : Equation (62) permits us to conclude
ðrÞ
that the two independent solutions of T~ n ðyÞ ¼ 0 (which are also solutions of Eq. (56) thanks
FOURTH-ORDER DIFFERENCE EQUATIONS
791
to Eq. (53)) are given by
C ðrÞ
n ðxÞ ¼ rðxÞQr21 ðxÞQnþr ðxÞ;
DðrÞ
n ðxÞ ¼ rðxÞQr21 ðxÞQnþr ðxÞ:
The four solutions of Eq. (56) obtained are linearly independent since Pn and Qn are two
linearly independent solutions of Eq. (5) and have different asymptotic behavior (see
Remark 1).
The proof of Eq. (58) already given in Ref. [9] uses the fact that since ðPn Þn and ðQn Þn
satisfy Eq. (1), each solution given in Eq. (57) satisfies the recurrence equation
X nþ1 ¼ ðx 2 bnþr ÞX n 2 gnþr X n21 ;
n $ 1:
ð63Þ
A
Remark 1 Following the method used in Ref. [28] (see p. 98), we get the asymptotic
formula for Qn ðzÞ in the discrete case
n
Q
Qn ðzÞ ¼ 2
i¼0
gi 1þO
rðzÞznþ1
1
;
z
provided that when z ! 1; the shortest distance from z to (a,b) is bounded away from zero.
The previous asymptotic formula can be used to deduce the asymptotic formula for the
solutions of the fourth-order difference equation give in Eq. (57).
n such that Pn and
If we replace the function of second kind Qn in Eqs. (57) and (58) by Q
21 ðxÞ ¼
Qn are two linearly independent solutions of Eq. (1) (with the initial condition Q
0 ðxÞ fixed) and Eq. (5), then the four linearly independent solutions of
2ð1=rðxÞÞ and Q
n : Also, the relation between PðrÞ
Eq. (56) are obtained just by replacing Qn in Eq. (57) by Q
n ;
Pn and Qn is obtained by replacing Qn in Eq. (58) by Qn ; however, the denominator g0 Gr of
r ðxÞ 2 Q
r21 ðxÞPr ðxÞÞ which is constant
Eq. (58) is to be replaced by the term rðxÞðPr21 ðxÞQ
with respect to x. This remark applies also for Theorems 3– 6.
Theorem 3 Let (Pn)n be a classical discrete orthogonal polynomial sequence, k [ N and
ðPn½k;m Þn the generalized co-recursive of (Pn)n. Four linearly independent solutions of the
difference equation
Fn½k;m ð yÞ ¼ 0;
satisfied by
ðPn½k;m Þn ;
n $ k þ 1;
ð64Þ
are (with n $ k þ 1)
An½k;m ðxÞ ¼ rðxÞP2k ðxÞPn ðxÞ;
Bn½k;m ðxÞ ¼ rðxÞP2k ðxÞQn ðxÞ;
ð65Þ
C n½k;m ðxÞ
¼ ½g0 Gk þ mrðxÞPk ðxÞ Qk ðxÞ Pn ðxÞ;
Dn½k;m ðxÞ ¼ ½g0 Gk þ mrðxÞ Pk ðxÞ Qk ðxÞ Qn ðxÞ;
where Qn is the function of second kind associated to (Pn)n defined by Eqs. (36) and (37).
792
M. FOUPOUAGNIGNI et al.
Moreover, Pn½k;m is related to these solutions by
Pn½k;m ¼
Proof
½g0 Gk þ mrðxÞPk ðxÞQk ðxÞPn ðxÞ 2 mrðxÞP2k ðxÞQn ðxÞ
;
g0 Gk
k $ 0; n $ k þ 1:
ð66Þ
By analogy with the proof of Theorem 2, we show using Eqs. (19), (43) and (48) that
Tn½k;m ðyÞ ¼ rðs þ tÞðsð1Þ þ tð1Þ ÞP2k ðxÞ P2k ðx þ 1ÞP2k ðx þ 2ÞDn ðzÞ=sð1Þ ;
where Tn½k;m is given by Eq. (50) and yðxÞ ¼ zðxÞrðxÞP2k ðxÞ: Therefore, An½k;m and Bn½k;m given
by
An½k;m ðxÞ ¼ rðxÞP2k ðxÞPn ðxÞ;
Bn½k;m ðxÞ ¼ rðxÞP2k ðxÞQn ðxÞ;
are two linearly independent solutions of
Tn½k;m ðyÞ ¼ 0:
Next, straightforward computation using Eqs. (19), (58) and (65) leads to
¼
P½k;m
n
C ½k;m
2 mB½k;m
n
n
;
g0 Gk
n $ k þ 1:
ð67Þ
m
Since the generalized co-dilated polynomials P½k;
and the function Bn½k;m given by Eq. (65),
n
are both solutions of the linear homogenous difference equation
F½k;m
ð yÞ ¼ 0;
n
n $ k þ 1;
it follows from Eq. (67) that the function C n½k;m ; given by Eq. (65), is also a solution of the
previous equation.
To prove that the function Dn½k;m is solution of Eq. (68), we proceed as follows:
In the first step, we write the expression Fn½k;m ðC n½k;m Þ in terms of Pn(x) and Pn(x þ 1) using
the first-order difference equation satisfied by the weight (see Eq. (3)) and the second-order
difference equation satisfied by Pn (see Eq. (5))
F½k;m
ðC ½k;m
ðxÞÞ ¼ G½k;m
ðxÞPn ðxÞ þ H ½k;m
ðxÞPn ðx þ 1Þ;
n
n
n
n
n $ k þ 1;
where Gn½k;m and H n½k;m are functions depending on r, s, t, Pk , Qk and ln.
m
In the second step, we use the fact that C½k;
is a solution of Eq. (64) and also the fact that
n
Pn(x) and Pn(x þ 1) are linearly independent to deduce that
¼ H ½k;m
¼ 0;
G½k;m
n
n
for n $ k þ 1:
In fact, assuming that Gn½k;m ðxÞ – 0; we get:
G½k;m
ðxÞPn ðxÞ þ H ½k;m
ðxÞPn ðx þ 1Þ ¼ 0¼)Pn ðxÞ ¼ 2
n
n
H ½k;m
ðxÞ
n
Pn ðx þ 1Þ:
½k;m
Gn ðxÞ
m
We deduce that G½k;
ðxÞ ¼ 2H n½k;m ðxÞ (since Pn is a monic polynomial of degree n).
n
We conclude that
m
0 ¼ Gn½k;m ðxÞPn ðxÞ þ H n½k;m ðxÞPn ðx þ 1Þ ¼ 2G½k;
ðxÞDðPn Þ;
n
n $ k þ 1:
FOURTH-ORDER DIFFERENCE EQUATIONS
793
The previous equation gives a contradiction because Gn½k;m – 0 and DðPn Þ – 0 (since
(D(Pn))n is orthogonal with respect to sðx þ 1Þrðx þ 1Þ [29]).
Finally, we use the fact that C n½k;m and Dn½k;m are multiples of Pn and Qn, respectively, with
the same multiplier factor namely g0 Gk þ mrPk Qk (see Eq. (65)), and the fact that Pn and Qn,
satisfy the same second-order difference equation (5) to get
F½k;m
ðD½k;m
ðxÞÞ ¼ G½k;m
ðxÞQn ðxÞ þ H ½k;m
ðxÞQn ðx þ 1Þ ¼ 0;
n
n
n
n
n $ k þ 1:
m
Therefore, D½k;
is also a solution of Eq. (64).
n
To complete the proof, we notice that An½k;m ; Bn½k;m ; C n½k;m and Dn½k;m are four linearly
independent solutions of Fn½k;m ðyÞ ¼ 0 since Pn and Qn are two linearly independent solutions
of Eq. (5) enjoying different asymptotic properties.
A
In the following, we give the equivalent of the previous theorem for the co-dilated classical
discrete orthogonal polynomials. The proof is similar to the one of the previous theorem by
using relations (26), (27), (43), (48) and (58).
Theorem 4 Let (Pn)n be a classical discrete orthogonal polynomial sequence, k [ N and
lj
ðPjk;
n Þn the generalized co-dilated of (Pn)n. Four linearly independent solutions of the
difference equation
mj
Fjk;
ð yÞ ¼ 0;
n
satisfied by
lj
ðPjk;
n Þn
n $ k þ 1;
ð68Þ
are (with n $ k þ 1)
lj
Ajk;
n ðxÞ ¼ rðxÞPk21 ðxÞPk ðxÞPn ðxÞ;
lj
Bjk;
n ðxÞ ¼ rðxÞPk21 ðxÞPk ðxÞQn ðxÞ;
ð69Þ
lj
C jk;
n ðxÞ ¼ ½g0 Gk þ ðl 2 1Þgk rðxÞ Pk21 ðxÞQk ðxÞPn ðxÞ;
lj
Djk;
n ðxÞ ¼ ½g0 Gk þ ðl 2 1Þgk rðxÞ Pk21 ðxÞQk ðxÞQn ðxÞ:
lj
The co-dilated Pjk;
is related to these solutions by
n
¼
Pjk;lj
n
½g0 Gk þ ðl 2 1Þgk rðxÞPk21 ðxÞQk ðxÞPn ðxÞ 2 ðl 2 1Þgk rðxÞPk21 ðxÞPk ðxÞQn ðxÞ
;
g0 Gk
ð70Þ
n $ k þ 1:
We furthermore, give the solutions for the generalized co-recursive associated and the
generalized co-modified classical orthogonal polynomials. The proofs are similar to the
previous ones.
Theorem 5 Let (Pn)n be a classical discrete orthogonal polynomial sequence, k [ N;
r [ N.0 and ðPn{r;k;m} Þn the generalized co-recursive associated with (Pn)n. Four linearly
independent solutions of the difference equation
Fn{r;k;m} ðyÞ ¼ 0;
n $ k þ 1;
ð71Þ
794
M. FOUPOUAGNIGNI et al.
satisfied by ðPn{r;k;m} Þn are (with n $ k þ 1)
m}
A{r;k;
ðxÞ ¼ ðg0 Gkþr Pr21 ðxÞ 2 mrðxÞPkþr ðxÞ½Pr21 ðxÞQkþr ðxÞ 2 Qr21 ðxÞPkþr ðxÞÞrðxÞPnþr ðxÞ;
n
m}
B{r;k;
ðxÞ ¼ ðg0 Gkþr Pr21 ðxÞ 2 mrðxÞPkþr ðxÞ½Pr1 ðxÞQkþr ðxÞ 2 Qr21 ðxÞ Pkþr ðxÞÞrðxÞQnþr ðxÞ;
n
C n{r;k;m} ðxÞ ¼ ðg0 Gkþr Qr21 ðxÞ 2 mrðxÞQkþr ðxÞ½Pr21 ðxÞQkþr ðxÞ 2 Qr21 ðxÞ Pkþr ðxÞÞrðxÞPnþr ðxÞ;
Dn{r;k;m} ðxÞ ¼ ðg0 Gkþr Qr21 ðxÞ 2 mrðxÞQkþr ðxÞ½Pr21 ðxÞQkþr ðxÞ 2 Qr21 ðxÞPkþr ðxÞÞrðxÞQnþr ðxÞ:
Moreover, Pn{r;k;m} is related to these solutions by
Pr21 ðxÞ mrðxÞPkþr ðxÞ ½Pr21 ðxÞQkþr ðxÞ 2 Qr21 ðxÞPkþr ðxÞ
P{r;k;m}
¼
2
rðxÞQnþr ðxÞ
n
g0 Gr21
g20 Gr21 Gkþr
2
Qr21 ðxÞ mrðxÞQkþr ðxÞ ½Pr21 ðxÞQkþr ðxÞ 2 Qr21 ðxÞPkþr ðxÞ
2
rðxÞPnþr ðxÞ;
g0 Gr21
g20 Gr21 Gkþr
ð72Þ
r $ 1; n $ k þ 1:
Theorem 6 Let (Pn)n be a classical orthogonal polynomial sequence, k [ N; and
ðPn½k;m;l Þn the generalized co-modified of (Pn)n. Four linearly independent solutions of the
difference equation
Fn½k;m;l ð yÞ ¼ 0; n $ k þ 1;
ð73Þ
m; l Þn are (with n $ k þ 1)
satisfied by ðP½k;
n
m; l ðxÞ ¼ ½ðl 2 1Þgk Pk21 ðxÞPk ðxÞ þ mP2k ðxÞrðxÞPn ðxÞ;
A½k;
n
m; l B½k;
ðxÞ ¼ ½ðl 2 1Þgk Pk21 ðxÞPk ðxÞ þ mP2k ðxÞrðxÞQn ðxÞ;
n
ð74Þ
Cn½k;m;l ðxÞ ¼ ½g0 Gk þ ðl 2 1Þgk rðxÞPk21 ðxÞQk ðxÞ þ mrðxÞPk ðxÞQk ðxÞPn ðxÞ;
m;l
ðxÞ ¼ ½g0 Gk þ ðl 2 1Þgk rðxÞPk21 ðxÞQk ðxÞ þ mrðxÞPk ðxÞQk ðxÞQn ðxÞ:
D½k;
n
The co-dilated Pn½k;m;l is related to these solutions by
ðl 2 1Þgk rðxÞPk21 ðxÞQk ðxÞ þ mrðxÞPk ðxÞQk ðxÞ
½k;m;l
Pn
¼ 1þ
Pn ðxÞ
g0 Gk
2
ðl 2 1Þgk rðxÞPk21 ðxÞPk ðxÞ þ mrðxÞP2k ðxÞ
Qn ðxÞ;
g0 Gk
n $ k þ 1:
ð75Þ
5. APPLICATIONS
5.1. On the rth Associated Charlier and Meixner Polynomials
ðrÞ
ðrÞ
For Charlier and Meixner polynomials, we give explicitly the operators SðrÞ
n ; Tn ; Fn and the
coefficient Xðs; t; Pr21 ; lr21 Þ: We also give the hypergeometric representation of the two
FOURTH-ORDER DIFFERENCE EQUATIONS
795
linearly independent solutions of Eq. (5) from which the hypergeometric representation of
the four solutions of the four-order difference equation FðrÞ
n ðyÞ ¼ 0 can be deduced.
5.1.1. The Charlier Case
The data for the Charlier polynomials cðaÞ
n ðxÞ (denoted in this paper by C n ðx; aÞ) involved in
Eqs. (1) –(5) are [17]:
sðxÞ ¼ x; tðxÞ ¼ a 2 x; ln ¼ n; rðxÞ ¼
ax
; x [ N; bn ¼ n þ a; gn ¼ n a; a . 0:
x!
The recurrence equation as well as the difference equation (see Eqs. (1) and (5)) satisfied
by the Charlier polynomials are given, respectively, by
aCnþ1 ðx;aÞ ¼ ðn þ a 2 xÞC n ðx;aÞ 2 nC n21 ðx;aÞ; n $ 1; C21 ðx;aÞ ¼ 0; C0 ðx;aÞ ¼ 1; ð76Þ
aC n ðx þ 1;aÞ þ ðn 2 x 2 aÞCn ðx;aÞ þ xC n ðx 2 1;aÞ ¼ 0
ð77Þ
The monic Charlier polynomial Pn(x) is related to the Charlier polynomial by
Pn ðxÞ ¼ ð2aÞn C n ðx;aÞ;
and satisfies the following normalized recurrence equation (see Eq. (1))
Pnþ1 ðxÞ ¼ ðx 2 n 2 aÞPn ðxÞ 2 anPn21 ðxÞ; n $ 1; P21 ðxÞ ¼ 0; P0 ðxÞ ¼ 1:
ð78Þ
The hypergeometeric representation of two linearly independent solutions of the
recurrence equations (76) and (77) are given by
!
2n; 2x 1
C n ðx; aÞ ¼2 F 0
;
ð79Þ
2
2 a
C n ðx; aÞ ¼
1
2F2
ðx þ 1Þðn þ 1Þ
n þ 2; x þ 2 1; 1
!
a :
ð80Þ
Remark 2
The polynomial C n ðx; aÞ given by Eq. (79) is the Charlier polynomial and satisfies
therefore Eqs. (76) and (77).
The function C n ðx; aÞ given by Eq. (80) satisfies also Eqs. (76) and (77). This can be
verified by using the command sumrecursion [15] which gives the recurrence
equation for sums of hypergeometric type.
Cn ðx; aÞ and C n ðx; aÞ are linearly independent solutions of Eq. (76) because the
Casorati determinant of these solutions of the second-order difference equation (76) given
796
M. FOUPOUAGNIGNI et al.
by (r(x) here is the Charlier weight)
W n ðx; aÞ ¼ Cn21 ðx; aÞC n ðx; aÞ 2 C n ðx; aÞC n21 ðx; aÞ
¼ GðnÞGðx þ 1Þ221 a xþn21
¼
GðnÞa 12n
2rðxÞ
is different from zero.
Cn ðx; aÞ and C n ðx; aÞ are linearly independent solutions of Eq. (77) because they remain
unchanged when we permutate the role of x and n, and the difference equation (77) is
obtained from Eq. (76) by permutation of x and n.
The difference operators are given by
FðrÞ ¼ aðn þ 2zÞ ðx þ 4ÞT4 þ ð22ax 2 4z 2 2z 3 þ 2n 2 2 6a þ 6z 2 2 3nz 2 2 n 2 z
þ 7nz 2 2nÞT3 þ ð2ax 2 5an þ 2z þ 4z 3 2 n 2 2 4zax 2 10za þ n 3 þ 4a 2 6z 2
þ 6nz 2 þ 4n 2 z 2 4nz 2 2axnÞT2 þ ð2ax þ 2z 2 2z 3 þ 4a 2 3nz 2 2 n 2 z þ nzÞT
þ aðn 2 2 þ 2zÞðx þ 1ÞI;
SðrÞ ¼ 2a 2 ðx þ 2ÞPr21 ðx þ 1Þð2n 2 2zÞðx þ 3ÞT2 þ ð2ðx þ 2Þðx þ 4Þðn 2 2 þ 2zÞ
£ ðn þ z þ 1Þðx þ 1ÞPr21 ðxÞ þ ðx þ 2Þðx þ 4Þð2ax þ 2z 2 2z 3 þ 4a 2 3nz 2 2 n 2 z þ nzÞ
£ Pr21 ðx þ 1ÞÞT þ a 2 ðx þ 2Þðn þ 2zÞðx þ 3ÞPr21 ðx þ 1ÞI;
TðrÞ ¼Pr21 ðx þ 1ÞPr21 ðxÞðx þ 2Þ2 aT2 þ ð2ðx þ 1Þðn þ z þ 1Þðx þ 2ÞPr21 ðxÞ2 2 z
£ ðn þ z þ 1Þðx þ 2ÞPr21 ðx þ 1ÞPr21 ðxÞÞT þ ð2aðx þ 1Þðx þ 2ÞPr21 ðx þ 1ÞPr21 ðxÞ
2 zaðx þ 2ÞPr21 ðx þ 1Þ2 ÞI:
Here, z is given by
z ¼ r 2 x 2 a 2 2;
and Pr21 represents the monic Charlier polynomial of degree r 2 1: The factor Xn is given by
X n ðs; t; Pr21 ; lr21 Þ ¼ 2ðx þ 2Þðx þ 3Þðx þ 1Þ2 ðx þ 4Þðz 2 1ÞPr21 ðx þ 1ÞP2r21 ðxÞ þ ðx þ 4Þ
ðx þ 3Þðx þ 2Þðx þ 1Þðax þ 2z þ 2a 2 2z 2 ÞP2r21 ðx þ 1ÞPr21 ðxÞ
þ ðx þ 2Þðx þ 3Þzðx þ 4Þðz þ 2a þ ax 2 z 2 ÞP3r21 ðx þ 1Þ:
FOURTH-ORDER DIFFERENCE EQUATIONS
797
5.1.2. The Monic Meixner Case
The data for the Meixner polynomials mðb;cÞ
n ðxÞ (denoted in this paper by M n ðx; b; cÞÞ are
[17]:
s ðxÞ ¼ x;
bn ¼
t ðxÞ ¼ ðc 2 1Þx þ bc;
n þ ðn þ bÞc
;
12c
gn ¼
ln ¼ ð1 2 cÞn;
nðn þ b 2 1Þc
;
ð1 2 cÞ2
rðxÞ ¼
ðbÞx c x
;
x!
x [ N;
b . 0; 0 , c , 1;
where (b)x represents the Pochhammer symbol defined by
ðbÞx ¼ bðb þ 1Þ . . . ðb þ x 2 1Þ;
x [ N; ðbÞ0 ; 1:
The recurrence equation as well as the difference equation (see Eqs. (1) and (5)) satisfied
by the Meixner polynomials are given, respectively, by
cðn þ bÞM nþ1 ðx; b; cÞ ¼ ðnðc þ 1Þ þ bc þ ðc 2 1ÞxÞM n ðx; b; cÞ 2 nM n21 ðx; b; cÞ; n $ 1;
M 21 ðx; b; cÞ ¼ 0;
ð81Þ
M 0 ðx; b; cÞ ¼ 1;
cðx þ bÞM n ðx þ 1;b;cÞ 2 ðð1 þ cÞx þ bc þ nðc 2 1ÞÞM n ðx;b;cÞ þ xM n ðx 2 1; b;cÞ ¼ 0: ð82Þ
The monic Meixner polynomial Pn(x) is related to the Meixner polynomial M n ðx; b;cÞ by
Pn ðxÞ ¼ ðbÞn
c n
M n ðx;b; cÞ;
c21
and satisfies the following normalized recurrence equation (see Eq. (1))
n þ ðn þ bÞc
nðn þ b 2 1Þc
Pnþ1 ðxÞ ¼ x 2
Pn21 ðxÞ;
Pn ðxÞ 2
12c
ð1 2 cÞ2
n $ 1; P21 ðxÞ ¼ 0; P0 ðxÞ ¼ 1:
ð83Þ
Hypergeometric representations of two linearly independent solutions of Eqs. (81) and
(82) are (with b – 1)
0
1
2n; 2x 1
1 2 A;
M n ðx; b; cÞ ¼2 F 1 @
c
b 1
0
1; 1; x þ n þ b þ 1 Gðx
þ
n
þ
b
þ
1ÞGðbÞ
c A:
@
n ðx; b; cÞ ¼
M
3 F2
bðx þ 1Þðn þ 1ÞGðx þ bÞGðn þ bÞ
x þ 2; n þ 2
ð84Þ
n ðx; b; cÞ are linearly independent
Remark 3 The proof of the fact that Mn ðx; b; cÞ and M
solutions of Eqs. (81) and (82) (for b – 1) is obtained following the way indicated in
n ðx; b; cÞ
Remark 2. In this case, the Casorati determinant of the solutions Mn ðx; b; cÞ and M
798
M. FOUPOUAGNIGNI et al.
given by
n ðx; b; cÞ 2 Mn ðx; b; cÞM
n21 ðx; b; cÞ
Zn ðx; b; cÞ ¼ Mn21 ðx; b; cÞM
¼
c12x2n ðb 2 1ÞGðbÞ2 GðnÞGðx þ 1Þ ðb 2 1ÞGðnÞc12n
¼
;
2Gðn þ bÞGðx þ bÞ
2ðbÞn rðxÞ
where r is the Meixner weight, vanishes only for b ¼ 1: Notice that the Casorati
determinants Wn ðx; aÞ for Charlier and Zn ðx; b; cÞ for Meixner cases were computed using
the relations they satisfy
n
Wnþ1 ðx; aÞ ¼ Wn ðx; aÞ;
a
Znþ1 ðx; b; cÞ ¼
n
Zn ðx; b; cÞ;
cðn þ bÞ
and the Maple command sumrecursion [15] in order to find the first-order difference
equations satisfied by W1 ðx; aÞ and Z1 ðx; b; cÞ:
n ðx; b; cÞ (see Eq. (84))
The function C n ðx; aÞ given by Eq. (80) can also be derived from M
using the following relation linking the Charlier and Meixner polynomials [17]
a
C n ðx; aÞ ¼ lim M n x; b;
:
b!1
aþb
n ðx; b; cÞ of Eqs. (76) and (77) (for Charlier)
Remark 4 The second solutions C n ðx; aÞ and M
and Eqs. (81) and (82) (for Meixner) given, respectively, by Eqs. (80) and (84) seem to be
new results. These hypergeometric representations are covergent and were obtained in the
following way: First, we neglect the first x þ 1 terms in the expression of Qn ðxÞ given by
1
Eq. (36) and get
X
rðsÞPn ðsÞ
n ðxÞ ¼ 1
:
Q
rðxÞ s¼xþ1 s þ x
Then we use the Maple command sumtohyper [15] to get the hypergeometric
n ðxÞ for the Charlier and Meixner polynomials. Finally, we remark that
representation of Q
n ðxÞ satisfies Eq. (1) and multiply it by an appropriate factor in order to ensure the
Q
n ðxÞ ¼ Q
x ðnÞ:
symmetry Q
The difference operators are given by
FðrÞ ¼ cð2z þ N 2 c 2 1Þðx þ 4Þðb þ x þ 3ÞT4 2 ð4 2 2bcx þ 2cN 2 þ 2N 2 2 24zc 2 2z 3
2 10xc þ 9z 2 þ 4c 3 þ 9zcN þ 9zN 2 zN 2 2 12zc 2 þ 9z 2 c 2 3z 2 N 2 2x 2 c 2
2 2bc 2 x 2 10xc 2 2 2x 2 c 2 6bc 2 2 6bc 2 6N 2 6c 2 N 2 12cN 2 12zÞT3
2 ð22 þ 4bcx 2 8xcN 2 5bcN 2 4cN 2 2 4N 2 þ N 3 þ 4zc þ 4z 3 þ 14xc 2 12z 2
2 10zbc 2 4zx 2 c 2 16zxc 2 4zbcx 2 2c 3 þ 6c 2 2 12zcN 2 12zN þ 4zN 2 þ 10zc 2
2 12z 2 c þ 6z 2 N þ 4x 2 c 2 þ 4bc 2 x þ 14xc 2 þ 4x 2 c þ 9bc 2 þ 6c þ 9bc þ 5N 2 2Nbcx
2 2Nx 2 c þ 5c 2 N þ 2cN þ 10zÞT2 2 ð4c 2 þ 6xc 2 þ 2x 2 c 2 þ 2bc 2 x þ 4bc 2
þ 3zcN þ 3z 2 c þ 4c þ 6xc þ 2x 2 c þ 2bcx þ 4bc þ 3zN þ 3z 2 2 3z 2 N
2 zN 2 2 2z 3 ÞT þ cð2z þ N 2 3c 2 3Þðx þ 1Þðb þ xÞI;
FOURTH-ORDER DIFFERENCE EQUATIONS
799
SðrÞ ¼ ðx þ 2Þðx þ 3Þð1 2 2z þ c 2 NÞðb þ x þ 2Þ ðb þ x þ 1Þ2 c 3 Pr21 ðx þ 1ÞT2
2 ððx þ 2Þðx þ 4Þcðb þ x þ 1Þðx þ 1ÞðN þ zÞð2N þ 3 2 2z þ 3cÞPr21 ðxÞ þ ðx þ 2Þðx þ 4Þ
£ cðb þ x þ 1Þð4c 2 þ 6xc 2 þ 2x 2 c 2 þ 2bc 2 x þ 4bc 2 þ 3zcN þ 3z 2 c þ 4c þ 6xc þ 2x 2 c
þ 2bcx þ 4bc þ 3zN þ 3z 2 2 3z 2 N 2 zN 2 2 2z 3 ÞPr21 ðx þ 1ÞÞT þ ðx þ 2Þðx þ 3Þ
£ ð1 2 2z þ c 2 NÞðb þ x þ 2Þðb þ x þ 1Þ2 c 3 Pr21 ðx þ 1ÞI;
TðrÞ ¼ cðx þ 2Þ2 ðb þ x þ 1ÞPr21 ðx þ 1ÞPr21 ðxÞT2 þ ð2ðx þ 2ÞðN þ zÞðx þ 1ÞPr21 ðxÞ2
2 zðN þ zÞðx þ 2ÞPr21 ðx þ 1ÞPr21 ðxÞÞT þ ð2cðb þ xÞðx þ 1Þðx þ 2ÞPr21 ðx þ 1ÞPr21 ðxÞ
2 ðx þ 2Þzcðb þ xÞPr21 ðx þ 1Þ2 ÞI:
Here N and z are given by
N ¼ ðn þ 1Þð1 2 cÞ;
z ¼ r 2 x 2 2 2 cðr þ x þ bÞ
and Pr21 is the monic Meixner polynomial of degree r 2 1: The expression Xn in this case is
given by
X n ; X n ðs; t; Pr21 ; lr21 Þ
¼ ðc 2 z þ 1Þðx þ 1Þ2 ðx þ 2Þðx þ 3Þðx þ 4ÞPr21 ðx þ 1ÞP2r21 ðxÞ þ ðx þ 1Þðx þ 2Þðx þ 3Þ
£ ðx þ 4Þðx 2 c þ 2zc þ bcx þ 3xc þ 2c þ 2bc þ 2z þ 2z 2 ÞP2r21 ðx þ 1ÞPr21 ðxÞ þ zðx þ 2Þ
£ ðx þ 3Þðx þ 4Þðx 2 c þ zc þ bcx þ 3xc þ 2c þ 2bc þ z 2 z 2 ÞP3r21 ðx þ 1Þ:
Notice that the difference operators FðrÞ
given for the rth associated
n
Charlier and Meixner polynomials coincide with those given in Ref. [19] with the
notations z ¼ R; r ¼ g:
5.2. Extension of Results to Real Order of Association
Let n be a real number with n $ 0 and ðPðrÞ
n Þn the family of polynomials defined by
ðnÞ
nÞ
Pnþ1
ðxÞ ¼ ðx 2 bnþn ÞPðnnÞ ðxÞ 2 gnþn Pðn21
ðxÞ;
n$1
ð85Þ
with the initial conditions
Pð0nÞ ðxÞ ¼ 1;
Pð1nÞ ðxÞ ¼ x 2 bv ;
where bnþn and gnþn are the coefficients bn and gn of Eq. (1) with n replaced by n þ n:
We assume that the starting family (Pn)n defined in (1) is classical discrete. The
coefficients bn and gn are therefore rational function in the variable n [16,18,28] and the
coefficients bnþn and gnþn well-defined. When gnþn – 0; ;n $ 1; the family ðPðrÞ
n Þn ; thanks
to Favard’s theorem [3,6] is orthogonal and represents the associated of the family ðPn Þn with
real order of association.
800
M. FOUPOUAGNIGNI et al.
Theorem 7 Let (Pn)n be a family of classical discrete orthogonal polynomial, n $ 0 a real
number and ðPðnnÞ Þn the n-associated of (Pn)n. We have:
1. ðPðnnÞ Þn satisfies
FðnnÞ ðyÞ ¼ 0;
ð86Þ
FðnnÞ
is the operator given in Eq. (52) with r replaced by n.
where
2. The difference operator FðnnÞ factorizes as
~ ðnÞ T
~ ðnÞ ¼ Xðs; t; Vn21 ; ln21 ÞFðnÞ ;
S
n
n
n
SðnnÞ TðnnÞ ¼ Xðs; t; Un21 ; ln21 ÞFðnnÞ ;
~ ðn Þ ;
S
n
ð87Þ
ðn Þ
~
T
and the factor X are those given in
where the operators SðnnÞ ; TðnnÞ ;
n
Eqs. (49) – (53) with r replaced by n, Pr and Qr are replaced by Un and Vn, respectively.
Un and Vn are the two linearly independent solutions of the difference equation
(see Ref. [28,29])
sðxÞD7yðxÞ þ tðxÞDyðxÞ þ ln yðxÞ ¼ 0;
ð88Þ
with Ur ¼ Pr ; Vr ¼ Qr for n ¼ r [ N and
n
ln ¼ 2 ððn 2 1Þs00 þ 2t0 Þ:
2
ð89Þ
Four linearly independent solutions of difference equation (86) are given by
AðnnÞ ðxÞ ¼ rðxÞUn21 ðxÞUnþn ðxÞ;
BðnnÞ ðxÞ ¼ rðxÞUn21 ðxÞVnþn ðxÞ;
ð90Þ
C nðnÞ ðxÞ ¼ rðxÞVn21 ðxÞUnþn ðxÞ;
DðnnÞ ðxÞ ¼ rðxÞVn21 ðxÞVnþn ðxÞ;
where r(x) is the weight function given by Eq. (3).
Proof
1. Let n be a fixed integer number and define the function F by
F : Rþ ! R n ! FðnnÞ ðPðnnÞ ðxÞÞ;
where Rþ is the set of positive real numbers. Using relation (85) for fixed x, F(n) can be
written as rational function in n. In fact, for the classical discrete orthogonal
polynomials, the three-term recurrence relation coefficients bn and gn are rational
functions in the variable n. Using Eq. (56) we get
ðrÞ
FðrÞ ¼ FðrÞ
n ðPn ðxÞÞ ¼ 0;
;r [ N:
We then conclude that FðnÞ is a rational function with an infinite number of zeros.
Therefore, FðnÞ ¼ 0; ;n [ Rþ and ðPðnnÞ Þn satisfies Eq. (86).
2. Equation (87) is proved by a straightforward computation using Un and Un which
satisfies Eq. (88).
FOURTH-ORDER DIFFERENCE EQUATIONS
801
3. The functions given in Eq. (90) are represented as products of functions satisfying
homogeneous difference equation of order 1 (for r) and 2 (for U and V). These functions,
therefore satisfy a difference equation of order 4 ð¼ 1 £ 2 £ 2Þ which is identical to
Eq. (86). Notice that by linear algebra one can deduce the difference equation of the
product (90), given the difference equations of the factors, since they have polynomial
coefficients. This can be done, e.g. by Maple command “rec*rec” [35] of the gfun
package.
We conclude the proof by noticing that the results of the previous theorem can be used
to extend Theorem 5 to the generalized co-recursive associated of classical discrete
orthogonal polynomials with real order of association as was done for classical continuous
in Ref. [9].
A
5.3. Solution of Some Second-order Difference Equations
The factorization pointed out in Eq. (51) can be used to prove the following:
Proposition 3
Two linearly independent solutions of the difference equation
SðrÞ
n ðyÞ ¼ 0;
are
ðrÞ
ðrÞ
EðrÞ
n ðxÞ ¼ Tn ðC n ðxÞÞ;
ðrÞ
ðrÞ
F ðrÞ
n ðxÞ ¼ Tn ðDn ðxÞÞ;
ðrÞ
where the operators SðrÞ
n and Tn are given by Eqs. (49) and (50), respectively, and the
ðrÞ
ðrÞ
functions C n ðxÞ and Dn ðxÞ given by Eq. (57).
Proposition 4
Two linearly independent solutions of the difference equation
SðnnÞ ðyÞ ¼ 0;
are
EðnnÞ ðxÞ ¼ TðnnÞ ðC ðnnÞ ðxÞÞ;
F ðnnÞ ðxÞ ¼ TðnnÞ ðDðnnÞ ðxÞÞ;
where the operators SðnnÞ and TðnnÞ are given by Eq. (87), and the functions CðnnÞ ðxÞ and DðnnÞ ðxÞ
given by Eq. (90).
ðrÞ
ðrÞ
Proof Since the functions C ðrÞ
n and Dn are solutions of equation Fn ðyÞ ¼ 0 (see Theorem 2),
we use the factorization given by Eq. (51) and get
ðrÞ
ðrÞ
SðrÞ
n ðTn ðyÞÞ ¼ Xðs; t; Pr21 ; lr21 ÞFn ðyÞ ¼ 0
ðrÞ
ðrÞ
ðrÞ
ðrÞ
for y [ {C ðrÞ
n ; Dn }: We therefore, conclude that the functions E n and F n satisfy Sn ðyÞ ¼ 0:
The proof of Proposition 4 is similar to the one of Proposition 3 by using Theorem 7. A
Remark 5 The previous propositions give solutions to families of second-order difference
equations. In particular, Proposition 3 solves a family of second-order difference equations
with polynomial coefficients. The two previous propositions, given for the associated
classical discrete orthogonal polynomials can be used to solve the difference equation
Sn ðyÞ ¼ 0 where Sn is the left factor of the factored form of the fourth-order difference
802
M. FOUPOUAGNIGNI et al.
operator Fn ðFn ¼ Sn Tn Þ for other modifications of classical discrete orthogonal polynomials
(see “Perturbation of recurrence coefficients” section).
5.4. Extension of Results to Semi-classical Cases
The proof of Theorem 1, which is the starting point of this paper, uses merely the secondorder difference equation (5) and the relation (32). Now we suppose that the family (Pn)n is
semi-classical discrete [7,14,22,25,26,34]. This implies that (Pn)n is orthogonal satisfying a
second-order difference equation of the form
n ðyðxÞÞ ¼ I 2 ðx; nÞyðx þ 2Þ þ I 1 ðx; nÞyðx þ 1Þ þ I 0 ðx; nÞyðxÞ ¼ 0;
M
ð91Þ
where the coefficients Ii(x,n)are polynomials in x of degree not depending on n.
For semi-classical orthogonal polynomials an equation of type (32) is known and can be
stated as [7,10]
~ n ðPð1Þ ðxÞÞ ¼ a1 ðxÞPn ðx þ 1Þ þ a0 ðxÞPn ðxÞ;
M
n21
ð92Þ
~ n a second-order linear difference operator with polynomial
where ai are polynomials and M
coefficients. Use of the two previous equations leads to the following extension.
Theorem 8 Given (Pn)n a sequence of semi-classical orthogonal polynomials satisfying
Eq. (91) and ðP n Þn a family of orthogonal polynomials obtained by modifying (Pn)n and
satisfying
P n ðxÞ ¼ An ðxÞPð1Þ
nþk21 þ Bn ðxÞPnþk ;
n $ k0 ;
ð93Þ
where An and Bn are polynomials of degree not depending on n, and k, k0 [ N; we have the
following:
1. The orthogonal polynomials ðP n Þn$k0 satisfy a common fourth-order linear difference
equation
F n ðyðxÞÞ ¼ K4 ðx; nÞyðx þ 4Þ þ K3 ðx; nÞyðx þ 3Þ þ K2 ðx; nÞyðx þ 2Þ þ K1 ðx; nÞyðx þ 1Þ
þ K0 ðx; nÞyðxÞ
¼ 0;
where the coefficients Ki are polynomials in x, with degree not depending on n.
2. The operator F n can be factored as product of two second-order linear difference
operators
n T n ;
F n ¼ S
n and T n are polynomials of degree not depending on n.
where the coefficients of S
The proof is similar to the one of Theorem 1 but with Eqs. (91) and (92) playing the role of
Eqs. (5) and (32), respectively.
The previous theorem covers many modifications of the recurrence coefficients of the
semi-classical discrete orthogonal polynomials, and in particular, the modifications such as
FOURTH-ORDER DIFFERENCE EQUATIONS
803
the associated, the general co-recursive, the general co-dilated, the general co-recursive
associated and the general co-modified semi-classical discrete orthogonal polynomials.
When the orthogonal polynomial sequence (Pn)n is semi-classical discrete, it is difficult in
n and T
~ n ; F n ; S
n; M
n in
general to represent the coefficients of the difference operators, M
terms of polynomials f and c, the coefficients of the functional equation (see Refs.
[7,13,22,34]) satisfied by the regular functional with respect to which ðP n Þn is orthogonal.
However, for particular cases (for example if the degrees of polynomials f and c are
small), it is possible after huge computations to give the coefficients of the difference
n and T
~ n ; F n ; S
n explicitly, and therefore look for functions annihilating
n; M
operators M
these difference operators.
Acknowledgements
We are very greateful to the Alexander von Humboldt Foundation, whose grant (Fellowship
No IV KAM 1070235), awarded to the first author Mama Foupouagnigni, was decisive for
the realization of this work.
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