Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
www.elsevier.com/locate/cam
Factorization of fourth-order di&erential equations for
perturbed classical orthogonal polynomials
M. Foupouagnignia;∗;1 , W. Koepf a , A. Ronveauxb
a
Department of Mathematics and Computer Science, University of Kassel, Heinrich-Plett Str. 40,
34132 Kassel, Germany
b
Facult%es Universitaires Notre Dame de la Paix, B-5000 Namur, Belgium
Received 20 February 2003; received in revised form 15 April 2003
Abstract
We factorize the fourth-order di&erential equations satis2ed by the Laguerre–Hahn orthogonal polynomials obtained from some perturbations of classical orthogonal polynomials such as: the rth associated (for
generic r), the general co-recursive, the general co-recursive associated, the general co-dilated and the general co-modi2ed classical orthogonal polynomials. Moreover, we 2nd four linearly independent solutions of
the fourth-order di&erential equations, and show that the factorization obtained for modi2cations of classical orthogonal polynomials is still valid, with some minor changes when the polynomial family modi2ed is
semi-classical. Finally, we extend the validity of the results obtained for the associated classical orthogonal
polynomials with integer order of association from integers to reals.
c 2003 Elsevier B.V. All rights reserved.
MSC: primary 33C45; secondary 33C47
Keywords: Classical and semi-classical orthogonal polynomials; Laguerre–Hahn class; Functions of the second kind;
Perturbed orthogonal polynomials; Second- and fourth-order di&erential equations
1. Introduction
Let U be a regular linear functional [8] on the linear space P of polynomials with real coe@cients
and (Pn )n a sequence of monic polynomials, orthogonal with respect to U, i.e.,
(i) Pn (x) = xn + lower degree terms,
∗
Corresponding author. Department of Mathematics, Advanced School of Education, University of Yaounde I, P.O.
Box 47, Yaounde, Cameroon.
E-mail addresses: [email protected], [email protected] (M. Foupouagnigni).
1
Research supported by the Alexander von Humboldt Foundation under grant no. IV KAM 1070235.
c 2003 Elsevier B.V. All rights reserved.
0377-0427/$ - see front matter doi:10.1016/j.cam.2003.04.005
300
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
(ii) U; Pn Pm = kn n; m ; kn = 0; n ∈ N,
where N = {0; 1; : : :} denotes the set of nonnegative integers. Here, ·; · means the duality bracket
and n; m the Kronecker symbol.
(Pn )n satis2es a three-term recurrence equation
Pn+1 (x) = (x − n )Pn (x) − n Pn−1 (x);
n¿1
(1)
with the initial conditions
P−1 (x) = 0;
P0 (x) = 1;
(2)
where n and n are real numbers with n = 0; ∀n ∈ N¿0 , and N¿0 denotes the set N¿0 = {1; 2; : : :}.
When the polynomial sequence (Pn )n is classical [28], i.e., orthogonal with respect to a positive
weight function (de2ned on the interval (a; b)) satisfying the 2rst-order di&erential equation called
Pearson equation:
() = (3)
with
xn (x)(x)|x=b
x=a = 0;
∀n ∈ N;
(4)
where is a polynomial of degree at most two and a 2rst-degree polynomial, each Pn satis2es
the di&erential equation
Ln (y(x)) = (x)y (x) + (x)y (x) + n y(x) = 0
(5)
and the orthogonality condition (ii) reads as
b
(x)Pn (x)Pm (x) d x = kn n; m ; kn = 0:
a
The coe@cients n ; n and n are given by [18,19]
n
n = − ((n − 1) + 2 ) = −n((n − 1)2 + 1 );
2
n =
−22 n2 1 − 21 1 n + 22 1 n + 22 0 − 1 0
;
(22 n − 22 + 1 )(22 n + 1 )
n = −n(1 − 22 + 2 n)(422 0 n2 − 822 0 n + 422 0 − 12 n2 2 + 212 n2
+ 42 0 n1 − 42 0 1 + 2 20 − 2 12 − 12 n1 − 1 1 0 + 0 21 + 12 1 )=
(22 n − 22 + 1 )2 (22 n − 2 + 1 )(22 n − 32 + 1 );
where
(x) = 2 x2 + 1 x + 0 ;
(x) = 1 x + 0
with
|1 |(|2 | + |1 | + |0 |) = 0:
The classical families are Jacobi, Laguerre and Hermite orthogonal polynomials [28].
(6)
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
301
Some modi2cations of Eq. (1) lead to new families of orthogonal polynomials such as the rth
associated (for generic r), the general co-recursive, the general co-recursive associated, the general
co-dilated and the general co-modi2ed classical orthogonal polynomials [23]. These new families of
orthogonal polynomials satisfy a common fourth-order linear homogeneous di&erential equation with
polynomial coe@cients of bounded degree. In general, they cannot satisfy a common second-order
linear homogeneous di&erential equation with polynomial coe@cients of bounded degree. Therefore,
these new polynomials are not semi-classical but belong to the Laguerre–Hahn class (see Section 2).
Many works have been devoted to the derivation of these fourth-order di&erential equations. Their
polynomial coe@cients have been given explicitly in [4,5,14,15,29,36,38] for the rth associated
classical orthogonal polynomials.
In 1994, using symbolic computation, the coe@cients of the fourth-order di&erential equation
for the co-recursive associated Laguerre and Jacobi orthogonal polynomials were given [20]. Also,
in [32], general fourth-order di&erential equation for the generalized co-recursive of all classical
orthogonal polynomials was given for any (but 2xed) level of recursivity using symbolic computation
software.
Despite the fact that apart from the rth associated orthogonal polynomials, the coe@cients of the
fourth-order di&erential equation satis2ed by the perturbed classical orthogonal polynomials require
heavy computations for being very large, we have succeeded in factorizing these fourth-order di&erential equations and also 2nding a basis of four linearly independent solutions of all the perturbed
systems of classical orthogonal polynomials considered. In Ref. [13], we succeeded also to factorize the equivalent fourth-order di&erence equation corresponding to the discrete case for which the
basic Eqs. (3) and (5) are di&erence equations of the same order, instead of di&erential equations.
Moreover, we have found interesting relations between the perturbed polynomials, the starting ones
and the functions of the second kind (see Section 2.2 for the de2nition).
In Section 2, we recall de2nitions and known results needed for this work. Section 3 is devoted to
the derivation and the factorization of the fourth-order di&erential equation. In Section 4, we solve
di&erential equations and represent perturbed classical orthogonal polynomials in terms of solutions
of second-order di&erential equations. In Section 5, we 2rst give asymptotic representation of solutions of the fourth-order di&erential equation for the rth associated classical orthogonal polynomials;
secondly, we extend the results obtained for the rth associated orthogonal polynomials with integer
order of association from integers to reals. Finally, we solve a family of second-order di&erential
equations and prove that the factorization obtained for modi2cations of classical orthogonal polynomials is still valid with some minor changes, when the polynomial family modi2ed is semi-classical
(see the next section for the de2nition).
2. Preliminaries and notations
In this section, we 2rst de2ne the semi-classical and the Laguerre–Hahn class of a given family
of orthogonal polynomials. Next, we present the families of rth associated, generalized co-recursive,
generalized co-dilated and generalized co-modi2ed orthogonal polynomials, and give relations between new sequences and the starting ones.
302
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
Each regular linear functional U generates a so-called Stieltjes function S of U de2ned by
U; xn ;
S(z) = −
n+1
z
n¿0
(7)
where U; xn are the moments of the functional U. The linear functional U satis2es in general a
simple functional equation living in P , the dual space of P. Appropriate de2nitions of (d=d x)(U)
and PU, where P is a polynomial allow to build a simple di&erential equation for the functional
which generalize in some way the Pearson equation for the weight [17] (see also [25,26]).
If the Stieltjes function S(x) satis2es a 2rst-order linear di&erential equation of the form
(x) S (x) = C(x)S(x) + D(x);
(8)
where ; C and D are polynomials, the functional U satis2es in P a 2rst-order di&erential equation
with polynomial coe@cients. In this case, the functional U and the corresponding orthogonal polynomial sequence (Pn )n belong to the semi-classical class (and are therefore called semi-classical)
which includes the classical families [3,17,25,26].
Each semi-classical orthogonal polynomial sequence (Pn )n satis2es a common second-order differential equation [17] (see also [25]).
Mn (y(x)) = I2 (x; n)y (x) + I1 (x; n)y (x) + I0 (x; n)y(x) = 0;
(9)
where Ii (x; n) are polynomials in x of degree not depending on n.
An important class, larger than the semi-classical one, appears when the Stieltjes function satis2es
a Riccati di&erential equation [9,11,22]
S = BS 2 + CS + D;
(10)
where = 0; B; C and D are polynomials. The corresponding functional U satis2es then a complicated quadratic di&erential equation in P . U and the corresponding orthogonal polynomials families
are said to belong to the Laguerre–Hahn class [9,11,22].
It is well known that any Laguerre–Hahn orthogonal polynomial sequence satis2es a common
fourth-order di&erential equation of the form [9,11,22]
J4 (x; n)y(4) (x) + J3 (x; n)y (x) + J2 (x; n)y (x) + J1 (x; n)y (x) + J0 (x; n)y(x) = 0;
where Ji (x; n) are polynomials of degree not depending on n.
It is shown, from several works [9–11,23,24] that 2nite perturbations of the recurrence coe@cients
of any semi-classical family generate orthogonal polynomials belonging to the Laguerre–Hahn and
therefore satisfy a fourth-order di&erential equation.
2.1. Perturbation of recurrence coe7cients
Now we consider a sequence of polynomials (Pn )n , orthogonal with respect to a regular linear
functional U, satisfying (1). Perturbations we will deal with are the following.
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
303
2.1.1. The rth associated orthogonal polynomials (Pn(r) )n
Given r ∈ N, the rth associated of the polynomials (Pn )n , is a polynomial sequence denoted by
(Pn(r) )n and de2ned by the recurrence equation (1) in which n and n are replaced by n+r and n+r ,
respectively,
(r)
Pn+1
(x) = (x − n+r )Pn(r) (x) − n+r Pn(r)
−1 (x);
n¿1
(11)
with the initial conditions
P0(r) (x) = 1;
P1(r) (x) = x − r :
(12)
The family (Pn(r) )n , thanks to Favard’s theorem [12], is orthogonal. It is related to the starting
polynomials and its 2rst associated by the relation [9]
Pn(r) (x) =
P (1) (x)
Pr −1 (x) (1)
Pn+r −1 (x) − r −2
Pn+r (x);
r − 1
r − 1
n ¿ 0; r ¿ 2;
(13)
where the sequence (n )n is de2ned by
n =
n
i ;
n ¿ 1; 0 ≡ 1:
(14)
i=1
2.1.2. The co-recursive (Pn[] )n and the generalized co-recursive orthogonal polynomials (Pn[k; ] )n
The co-recursive of the orthogonal polynomial sequence (Pn )n , denoted by (Pn[] )n , was introduced
for the 2rst time by Chihara [7], as the family of polynomials generated by the recursion formula
(1) in which 0 is replaced by 0 + []
Pn+1
(x) = (x − n )Pn[] (x) − n Pn[]
−1 (x);
n¿1
(15)
with the initial conditions
P0[] (x) = 1;
P1[] (x) = x − 0 − ;
(16)
where denotes a real number.
This notion was extended to the generalized co-recursive orthogonal polynomials in [9,10,30] by
modifying the sequence (
n )n at the level k. This yields an orthogonal polynomial sequence denoted
by (Pn[k; ] )n and generated by the recursion formula
[k; ]
Pn+1
(x) = (x − n∗ )Pn[k; ] (x) − n Pn[k;−]1 (x);
n¿1
(17)
with the initial conditions
P0[k; ] (x) = 1;
P1[k; ] (x) = x − 0∗ ;
where n∗ = n for n = k and k∗ = k + .
(18)
304
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
The orthogonal polynomial sequence (Pn[k; ] )n is related to (Pn )n and it is associated by [23]
Pn[k; ] (x) = Pn (x) − Pk (x)Pn(k+1)
−(k+1) (x);
Pn[k; ] (x) = Pn (x);
n ¿ k + 1;
n 6 k:
(19)
Use of (13) transforms the previous equations in
(1)
2
P
(x)P
P
(x)
k
(1)
−
1
k
Pn (x);
Pn−1 (x) + 1 +
Pn[k; ] (x) = − k
k
k
Pn[k; ] (x) = Pn (x);
n ¿ k + 1;
n 6 k:
(20)
Obviously, we have the relations Pn[0; ] = Pn[] , and Pn[0] = Pn .
{r; }
2.1.3. The co-recursive associated (Pn )n and the generalized co-recursive associated orthogonal
{r; k; }
)n
polynomials (Pn
The co-recursive associated as well as the generalized co-recursive associated of the orthogonal
{r; }
{r; k; }
)n , respectively, are, the co-recursive
polynomial sequence (Pn )n , denoted by (Pn )n and (Pn
and the generalized co-recursive (with modi2cation on k ) of the associated (Pn(r) )n of (Pn )n , respectively. Thanks to (19), they are related with (Pn )n and it is associated by
Pn{r; 0; } = Pn{r; }
and
Pn{r; k; } (x) = Pn(r) (x) − Pk(r) (x)Pn(r+k+1)
−(k+1) (x);
Pn{r; k; } (x) = Pn(r) (x);
n ¿ k + 1;
n 6 k:
(21)
The generalized co-recursive associated orthogonal polynomials can also be expressed using (13)
and (21) by
(r)
Pr −1 (x) Pk+r (x)Pk (x)
(1)
{r; k; }
Pn+r
Pn
(x) =
−
−1 (x)
r+k
r − 1
(1)
(1)
(r)
Pr −2 (x) Pk+r −1 (x)Pk (x)
−
Pn+r (x); n ¿ k + 1;
−
r − 1
r+k
Pn{r; k; } (x) = Pn (x)(r) ;
| |
n 6 k:
(22)
|k; |
2.1.4. The co-dilated (Pn )n and the generalized co-dilated orthogonal polynomials (Pn )n
| |
The co-dilated of the orthogonal polynomial sequence (Pn )n , denoted by (Pn )n , was introduced
by Dini [9], as the family of polynomials generated by the recursion formula (1) in which 1 ,
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
305
is replaced by 1 , i.e.,
| |
| |
Pn+1 (x) = (x − n )Pn|| (x) − n Pn−1 (x);
n¿2
(23)
with the initial conditions
| |
P0 (x) = 1;
| |
P1 (x) = x − 0 ;
| |
P2 (x) = (x − 0 )(x − 1 ) − 1 ;
(24)
where is a nonzero real number.
This notion was extended to the generalized co-dilated orthogonal polynomials in [10,30] by
modifying the sequence (n )n at the level k. This yields an orthogonal polynomial sequence denoted
|k; |
by (Pn )n and generated by the recurrence equation
|k; |
|k; |
Pn+1 (x) = (x − n )Pn|k; | (x) − ∗n Pn−1 (x);
n¿1
(25)
with the initial conditions
|k; |
P0
(x) = 1;
|k; |
P1
(x) = x − 0 ;
(26)
where ∗n = n for n = k and ∗k = k .
|k; |
The orthogonal polynomial sequence (Pn )n is related to (Pn )n and its associated by [23]
Pn|k; | (x) = Pn (x) + (1 − )k Pk −1 (x)Pn(k+1)
−(k+1) (x);
Pn|k; | (x) = Pn (x);
n ¿ k + 1;
n 6 k:
(27)
Use of (13) transforms the previous equation in
(1)
(1
−
)P
(x)P
(x)
(1 − )Pk −1 (x)Pk (x) (1)
k
−
1
k −1
Pn (x) +
Pn−1 (x);
Pn|k; | (x) = 1 −
k − 1
k − 1
Pn|k; | (x) = Pn (x);
n 6 k:
n ¿ k + 1;
(28)
For k = 1 or = 1, we have
Pn|1; | = Pn|| ;
Pn|k; 1| = Pn :
2.1.5. The generalized co-modi:ed orthogonal polynomials (Pn[k; ; ] )n
New families of orthogonal polynomials can also be generated by modifying at the same time the
sequences (
n )n and (n )n at the levels k and k , respectively. When k = k , the new family obtained
[23], denoted by (Pn[k; ; ] )n is generated by the three-term recurrence relation
[k; ; ]
Pn+1
(x) = (x − n∗ )Pn[k; ; ] (x) − ∗n Pn[k;−;1 ] (x);
n¿1
(29)
306
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
with the initial conditions
P0[k; ; ] (x) = 1;
P1[k; ; ] (x) = x − 0∗ ;
(30)
where n∗ = n ; ∗n = n for n = k and k∗ = k + ; ∗k = k . This family is represented in terms of
the starting polynomials and their associated ones by [23]
Pn[k; ; ] (x) = Pn (x) + ((1 − )k Pk −1 (x) − Pk (x))Pn(k+1)
−(k+1) (x);
Pn|k; | (x) = Pn (x);
n ¿ k + 1;
n 6 k:
(31)
The latter relation can also be written as
(1 − )Pk −1 (x)Pk(1)
Pk (x)Pk(1)
−1 (x)
−1 (x)
[k; ; ]
Pn
Pn (x)
(x) = 1 −
+
k − 1
k
(1 − )Pk −1 (x)Pk (x) Pk2 (x)
+
Pn(1)
−
n ¿ k + 1;
−1 (x);
k − 1
k
Pn[k; ] (x) = Pn (x);
n 6 k:
(32)
2.2. Results on classical orthogonal polynomials
Next, we state the following lemmas which are essential for this work.
Lemma 1 (Ronveaux [29]): Given a classical orthogonal polynomial sequence (Pn )n satisfying (5),
the following relation holds:
Ln∗ (Pn(1)
−1 (x)) = ( − 2 )Pn (x);
(33)
where Ln∗ , which is the adjoint of Ln is given by
Ln∗ = D2 + (2 − )D + (n + − ):
(34)
It should be noticed that Ln and Ln∗ are related by
Ln (y) = Ln∗ (y);
∀y;
(35)
where is the weight function satisfying Eqs. (3) and (4).
Lemma 2 (Nikiforov and Uvarov [28]). (1) Two linearly independent solutions of the di>erential
equation
Ln (y(x)) = (x)y (x) + (x)y (x) + n y(x) = 0;
are Pn and Qn , where (Pn )n is the polynomial sequence, orthogonal with respect to the weight
function de:ned on the interval (a; b), satisfying Eqs. (3) and (4). The constant n
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
307
is given by
n
n = − ((n − 1) + 2 ):
2
Qn is the function of the second kind, de:ned by
b
(s)Pn (s)
1
ds:
Qn (x) =
(x) a
s−x
(36)
(2) The polynomials Pn and the function Qn are two linearly independent solutions of the recurrence
equation (1).
Notice that the representation of Qn given above is valid for x ∈ [a; b]. But this representation
is still valid for x ∈ [a; b] by analytic continuation [28] or by taking Cauchy’s principal part in the
integral of (36) [14]. Pn and Qn are given for each classical situation in terms of hypergeometric
functions (see Section 5).
3. Factorization of fourth-order dierential operators
Given (Pn )n a classical orthogonal polynomial sequence, we consider in general all transformations
which lead to new families of orthogonal polynomials denoted by (PO n )n and are related to the starting
sequence by
(1)
PO n (x) = An (x)Pn+k
−1 + Bn (x)Pn+k ;
n ¿ k ;
(37)
where An and Bn are polynomials of degree not depending on n, and k; k ∈ N. We have the following:
Theorem 1. (1) The orthogonal polynomials (PO n )n¿k satisfy a common fourth-order linear di>erential equation
Fn (y(x)) = J4 (x; n)y (x) + J3 (x; n)y (x) + J2 (x; n)y (x)
+ J1 (x; n)y (x) + J0 (x; n)y(x) = 0;
(38)
where the coe7cients 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 di>erential operators
Sn and Tn
Fn = S n Tn ;
(39)
where the coe7cients in Sn and Tn are polynomials of degree not depending on n.
(1)
Proof. In the 2rst step, we solve Eq. (37) in terms of Pn+k
−1
(1)
Pn+k
−1 (x) =
PO n (x) − Bn (x)Pn+k (x)
An (x)
(40)
308
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
and substitute the previous relation in Eq. (33) in which n is replaced by n + k. Then we use (5)
(x) and get
(for Pn+k ) to eliminate the term Pn+k
+ b0 Pn+k ;
Mn+k (PO n ) = b1 Pn+k
(41)
where bi are rational functions and Mn+k a second-order linear operator given in terms of operator
∗
Ln+k
(see (33)) by
y
3 ∗
:
(42)
Mn+k (y) = An Ln+k
An
(x), and get
Next, we take derivative in (41) and use again (5) to eliminate Pn+k
+ c0 Pn+k :
[Mn+k (PO n )] = c1 Pn+k
(43)
We reiterate the same process using the previous equation and get
[Mn+k (PO n )] = d1 Pn+k
+ d0 Pn+k ;
(44)
where ci and di are again rational functions.
The fourth-order di&erential equation is given in determinantal form from (41), (43) and (44)
b1 b0
Mn+k (PO n ) (45)
Fn (PO n ) = c1 c0 [Mn+k (PO n )] = 0:
d1 d0 [Mn+k (PO n )] The previous equation can be written as
Fn (PO n ) = e2 [Mn+k (PO n )] + e1 [Mn+k (PO n )] + e0 Mn+k (PO n ) = [Sn Tn ](PO n ) = 0;
(46)
where the second-order di&erential operators Sn and Tn are given by
Sn = e2 D2 + e1 D + e0 ;
Tn = Mn+k :
(47)
We conclude the proof by noticing that after cancellation of the denominator in (45), the coe@cients
ei are polynomials of degree not depending on n.
It should be mentioned that the previous method was 2rst developed in [4]. The more general
situation considered in [23] gives the fourth-order di&erential equation for the orthogonal polynomial
sequence in the form
P̃ n = Pn + QPn(k+1)
−(k+1) ;
where Q is a polynomial of degree k and (Pn )n a semi-classical orthogonal polynomial sequence. The
previous theorem (also valid for semi-classical orthogonal polynomials (see Section 5.5)) therefore
extends the results given in [23]. In addition, we would like to mention that the factorization pointed
out in the previous theorem (except some particular cases listed below) seems to be a new result
and has many applications as will be shown later. This factorization was known for some particular
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
309
cases: the case of the 2rst associated classical orthogonal polynomials (see [29]), which is obviously
a consequence of (33) and the case of co-recursive orthogonal polynomials given explicitly in [31].
{r; k; } |k; |
Fn and Fn[k; ; ] the fourth-order di&erential
In what follows, we will denote by Fn(r) ; Fn[k; ] ; Fn
operators (obtained after cancellation of common factors) for the rth associated, the generalized
co-recursive, the generalized co-recursive associated, the generalized co-dilated, and the generalized
co-modi2ed orthogonal polynomials.
3.1. Some consequences
For the rth associated classical orthogonal polynomials (Pn(r) )n , we have used the previous theorem
and the representation given in (13) to compute the operators Sn and Tn using Maple 7 [27].
Proposition 1. The two di>erential operator factors of the fourth-order di>erential operator for
the rth associated classical orthogonal polynomials are
Sn(r) = Pr −1 D2 + [( + )Pr −1 − 2Pr −1 ]D + [( + n+r − r −1 )Pr −1 − 2Pr −1 ];
(48)
Tn(r) = Pr2−1 D2 − Pr −1 [( − 2 )Pr −1 + 2Pr −1 ]D
+ [2( − )Pr −1 Pr −1 + 2Pr −1 Pr −1 + ( − + n+r + r −1 )Pr2−1 ];
(49)
where r ∈ N¿0 and (Pn )n is the sequence of classical orthogonal polynomials satisfying (5).
Moreover, we have
Sn(r) Tn(r) = Pr3−1 Fn(r) ;
(50)
where
2
Fn(r) = 2 D4 + 5 D3 + (6 − 2 + 2 + 2n+r + 2r −1 − 2 + 3 )D2
+ 3(r −1 + n+r − + + )D
2
+ [(n+r − r −1 )2 + (n+r + r −1 ) + − ]:
(51)
Remark 1. It should be mentioned that the factorization pointed out in (50) for generic r was already
known for the 2rst associated (r = 1) [29]:
Fn(1) = (D2 + ( + )D + + n+1 )(D2 + (2 − )D + − + n+1 ):
(52)
Eq. (51) gives a new representation of the fourth-order di&erential equation for the rth associated
classical orthogonal polynomials in terms of ; and n and of course can be brought in the
form of known results [36,5,38]. From this representation, we recover easily the simple form of the
fourth-order di&erential equation for rth associated classical orthogonal polynomials, given in [21]
Fn(r) = Fn(1) + (1 − r)[(n + r − 2) + 2 ][2D2 + 3 D − (n2 − 1) ];
where Fn(1) is given by (52).
310
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
Corollary 1. The fourth-order di>erential operator can also be factorized as
S̃n(r) T̃n(r) = Qr3−1 Fn(r) ;
(53)
where the operators S̃n(r) and T̃n(r) are obtained from the operators Sn(r) and Tn(r) , respectively by
replacing the polynomials Pr −1 with the function Qr −1 , i.e.,
S̃n(r) = Qr −1 D2 + [( + )Qr −1 − 2Qr −1 ]D + [( + n+r − r −1 )Qr −1 − 2Qr −1 ];
(54)
T̃n(r) = Qr2−1 D2 − Qr −1 [( − 2 )Qr −1 + 2Qr −1 ]D
+ [2( − )Qr −1 Qr −1 + 2Qr −1 Qr −1 + ( − + n+r + r −1 )Qr2−1 ]:
(55)
The proof is obtained by computation utilizing the fact that Qn satis2es (5).
Proposition 2. The operator Tn for the generalized co-recursive and co-dilated classical orthogonal
|k; |
|k; |
]
; Tn are obtained
polynomials (Pn[k; ] )n and (Pn )n (with k ¿ 1), denoted, respectively, by T[k;
n
in the same way:
]
T[k;
= Pk2 D2 − Pk [( − 2 )Pk + 4Pk ]D
n
+ [4( − )Pk Pk + 6Pk Pk + (n + 2k + − )Pk2 ];
(56)
T|nk; | = Pk2−1 Pk2 − Pk −1 Pk [2(Pk −1 Pk ) + ( − 2 )Pk −1 Pk ]D
+ [(k −1 + k + n + − )Pk2−1 Pk2 + ( − )(Pk2−1 Pk2 )
+ 2Pk −1 Pk −1 Pk2 + 2Pk2−1 Pk Pk + 2Pk −1 Pk Pk −1 Pk ]:
(57)
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. (22)
and (32). The same remark applies for the factors Sn and Tn of the fourth-order di&erential equation
satis2ed by the generalized co-recursive associated and co-modi2ed classical orthogonal polynomials.
4. Solutions of the fourth-order dierential equations
In the following, we solve the fourth-order di&erential equation satis2ed by the 2ve perturbations
listed in Section 2 and represent the new families of orthogonal polynomials in terms of solutions
of second-order di&erential equations.
Theorem 2. Let (Pn )n be a classical orthogonal polynomial sequence, r ∈ N¿0 and (Pn(r) )n the rth
associated of (Pn )n . Four linearly independent solutions of the di>erential equation
Fn(r) (y) = 0
(58)
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
311
satis:ed by (Pn(r) )n , where Fn(r) is given by (51), are:
An(r) (x) = (x)Pr −1 (x)Pn+r (x);
Bn(r) (x) = (x)Pr −1 (x)Qn+r (x);
(59)
Cn(r) (x) = (x)Qr −1 (x)Pn+r (x);
Dn(r) (x) = (x)Qr −1 (x)Qn+r (x);
Qn denoting the function of second kind associated to (Pn )n which is de:ned by (36).
Moreover, Pn(r) is related to these solutions by
Pn(r) (x) =
=
Bn(r) (x) − Cn(r) (x)
0 r − 1
(x)(Pr −1 (x)Qn+r (x) − Qr −1 (x)Pn+r (x))
;
0 r − 1
where k is given by (14) and 0 de:ned as
b
0 =
(x) d x:
a
∀n ∈ N; ∀r ∈ N¿0 ;
(60)
(61)
Proof. In the 2rst step, we solve the di&erential equation
Tn(r) (y) = 0:
To do this, we use (35), (42) and (47) to get
Tn(r) (y) = Mn+r (y)
∗
= Pr3−1 Ln+r
y
Pr − 1
= Pr3−1 Ln+r (z);
(62)
where y = zPr −1 . Since the two linearly independent solutions of Ln+r (z) = 0 are Pn+r and Qn+r
(see Lemma 2), the two linearly independent solutions of Tn(r) (y) = 0 (which are also solutions of
(58) thanks to (50)) are
An(r) (x) = (x)Pr −1 (x)Pn+r (x);
Bn(r) (x) = (x)Pr −1 (x)Qn+r (x):
(63)
Use of (55) utilizing the fact that the weight function and the function Qn satisfy (3) and (5),
respectively, leads to
T̃n(r) (y) = Qr3−1 Ln+r (z);
(64)
312
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
where y = zQr −1 . Eq. (64) permits us to conclude that the two independent solutions of T̃n(r) (y) = 0
(which are also solutions of (58) thanks to (53)) are given by
Cn(r) (x) = (x)Qr −1 (x)Pn+r (x);
Dn(r) (x) = (x)Qr −1 (x)Qn+r (x):
The four solutions of (58) obtained are linearly independent since Pn and Qn are two linearly
independent solutions of (5) and have di&erent asymptotic behaviour (see Section 5.1).
It should be mentioned that computations with Maple 7 using the fact that Pn and Qn satisfy (5),
con2rm that the functions An(r) ; Bn(r) ; Cn(r) and Dn(r) satisfy (58). Also, notice that the structure of the
solutions of Eq. (58) given by (59) was suggested by Hahn [16].
To prove (60) one has to remark that since (Pn )n and (Qn )n satisfy (1), each solution given in
(59) satis2es the recurrence equation
Xn+1 = (x − n+r )Xn − n+r Xn−1 ;
n ¿ 1:
(65)
Therefore, the function Xn(r) de2ned by
Xn(r) (x) =
(x)(Pr −1 (x)Qn+r (x) − Qr −1 (x)Pn+r (x))
;
0 r − 1
r ¿ 1;
ful2lls (65). It remains to prove that the initial values are X0(r) = 1; X1(r) = x − r . We have
X0(r+1) =
(x)(Pr (x)Qr+1 (x) − Qr (x)Pr+1 (x))
0 r
=
(x)(Pr (x)[(x − r )Qr (x) − r Qr −1 (x)] − Qr (x)[(x − r )Pr (x) − r Pr −1 (x)])
0 r
=
(x)(Pr −1 (x)Qr (x) − Qr −1 (x)Pr (x))
0 r − 1
= X0(r) :
We deduce that
X0(r) = X0(1) ;
r ¿ 1:
A computation using (36) and (61) gives
(x)(P0 (x)Q1 (x) − Q0 (x)P1 (x))
0
b
1
(s − 0 )(s) ds (x − 0 ) b (s) ds
−
=
0 a
s−x
0
s−x
a
X0(1) =
= 1:
Therefore,
X0(r) = 1;
r ¿ 1:
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
313
Use of (1) for Pr+1 and Qr+1 and the previous equation gives
X1(r) = (x − r )X0(r) = x − r :
We therefore conclude that
Xn(r) = Pn(r) ;
r ¿ 1; n ¿ 0;
since (Pn(r) )n is the unique solution of recurrence equation (65) with the initial conditions
P0(r) = 1;
P1(r) = x − r :
Remark 2. The results of the previous theorem are still valid if we replace Pn and Qn in Eqs. (59)
and (60) by two other linearly independent solutions of Eqs. (1) and (5). In fact, the structure of
the solutions given in (59) remains the same. The same remark applies for (60) except that the
denominator in (60) may be a di&erent constant (with respect to x) factor.
Theorem 3. Let (Pn )n be a classical orthogonal polynomial sequence, k ∈ N and (Pn[k; ] )n the generalized co-recursive of (Pn )n . Four linearly independent solutions of the di>erential equation
Fn[k; ] (y) = 0;
n¿k + 1
(66)
satis:ed by (Pn[k; ] )n , are (with n ¿ k + 1)
]
(x) = (x)Pk2 (x)Pn (x);
A[k;
n
Bn[k; ] (x) = (x)Pk2 (x)Qn (x);
(67)
Cn[k; ] (x) = [0 k + (x)Pk (x)Qk (x)]Pn (x);
Dn[k; ] (x) = [0 k + (x)Pk (x)Qk (x)]Qn (x);
where Qn is the function of second kind associated to (Pn )n de:ned by (36).
Moreover, Pn[k; ] is related to these solutions by
Pn[k; ] =
[0 k + (x)Pk (x)Qk (x)]Pn (x) − (x)Pk2 (x)Qn (x)
;
0 k
k ¿ 0; n ¿ k + 1:
Proof. By analogy with the proof of Theorem 2, we show using (20), (42) and (47) that
]
(y) = Pk6 Ln (z);
T[k;
n
]
]
where T[k;
is given by (56) and y = zPk2 . Therefore, A[k;
and Bn[k; ] given by
n
n
]
(x) = (x)Pk2 (x)Pn (x);
A[k;
n
Bn[k; ] (x) = (x)Pk2 (x)Qn (x);
are two linearly independent solutions of
]
(y) = 0:
T[k;
n
(68)
314
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
Next, we use (19) and (60) and get
Pn[k; ] =
Cn[k; ] − Bn[k; ]
;
0 k
n ¿ k + 1:
(69)
Since the generalized co-dilated polynomials Pn[k; ] and the function Bn[k; ] given by (67), are both
solutions of the linear homogeneous di&erential equation
Fn[k; ] (y) = 0;
n ¿ k + 1;
it follows from (69) that the function Cn[k; ] , given by (67), is also solution of the previous equation.
Computations show that the function Dn[k; ] , given by (67) is also solution of the previous di&erential
equation. One can also prove that Dn[k; ] is solution of the previous di&erential equation by following
the proof given in [13] for the discrete case.
]
To complete the proof, we notice that A[k;
; Bn[k; ] ; Cn[k; ] and Cn[k; ] are four linearly independent
n
[k; ]
solutions of Fn (y) = 0 since Pn and Qn are two linearly independent solutions of (5) enjoying
di&erent asymptotic properties.
In the following, we give the equivalent of the previous theorem for the co-dilated classical
orthogonal polynomials. The proof is similar to the one of the previous theorem by using relations
(27), (28), (42), (47), and (60).
|k; |
Theorem 4. Let (Pn )n be a classical orthogonal polynomial sequence, k ∈ N and (Pn )n the generalized co-dilated of (Pn )n . Four linearly independent solutions of the di>erential equation
Fn|k; | (y) = 0;
|k; |
satis:ed by (Pn
n ¿ k + 1;
(70)
)n are (with n ¿ k + 1)
A|nk; | (x) = (x)Pk −1 (x)Pk (x)Pn (x);
Bn|k; | (x) = (x)Pk −1 (x)Pk (x)Qn (x);
Cn|k; | (x) = [0 k + ( − 1)k (x)Pk −1 (x)Qk (x)]Pn (x);
(71)
Dn|k; | (x) = [0 k + ( − 1)k (x)Pk −1 (x)Qk (x)]Qn (x):
|k; |
The co-dilated Pn
Pn|k; | =
is related to these solutions by
[0 k + ( − 1)k (x)Pk −1 (x)Qk (x)]Pn (x) − ( − 1)k (x)Pk −1 (x)Pk (x)Qn (x)
;
0 k
n ¿ k + 1:
(72)
We furthermore give the solutions for the generalized co-recursive associated and the generalized
co-modi2ed classical orthogonal polynomials. The proofs are similar to the previous ones.
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
315
{r; k; }
Theorem 5. Let (Pn )n be a classical orthogonal polynomial sequence, k ∈ N; r ∈ N¿0 and (Pn
)n
the generalized co-recursive associated of (Pn )n . Four linearly independent solutions of the di>erential equation
Fn{r; k; } (y) = 0;
{r; k; }
satis:ed by (Pn
n ¿ k + 1;
(73)
)n are (with n ¿ k + 1)
A{n r; k; } (x) = (0 k+r Pr −1 (x) − (x)Pk+r (x)[Pr −1 (x)Qk+r (x) − Qr −1 (x)Pk+r (x)])(x); Pn+r (x);
Bn{r; k; } (x) = (0 k+r Pr −1 (x) − (x)Pk+r (x)[Pr −1 (x)Qk+r (x) − Qr −1 (x)Pk+r (x)])(x)Qn+r (x);
Cn{r; k; } (x) = (0 k+r Qr −1 (x) − (x)Qk+r (x)[Pr −1 (x)Qk+r (x) − Qr −1 (x)Pk+r (x)])(x)Pn+r (x);
Dn{r; k; } (x) = (0 k+r Qr −1 (x) − (x)Qk+r (x)[Pr −1 (x)Qk+r (x) − Qr −1 (x)Pk+r (x)])(x)Qn+r (x):
{r; k; }
Moreover, Pn
is related to these solutions by
Pr −1 (x) (x)Pk+r (x)[Pr −1 (x)Qk+r (x) − Qr −1 (x)Pk+r (x)]
{r; k; }
(x)Qn+r (x)
=
−
Pn
0 r − 1
20 r −1 k+r
Qr −1 (x) (x)Qk+r (x)[Pr −1 (x)Qk+r (x) − Qr −1 (x)Pk+r (x)]
(x)Pn+r (x);
−
−
0 r − 1
20 r −1 k+r
r ¿ 1; n ¿ k + 1:
(74)
Theorem 6. Let (Pn )n be a classical orthogonal polynomial sequence, k ∈ N and (Pn[k; ; ] )n the
generalized co-modi:ed of (Pn )n . Four linearly independent solutions of the di>erential equation
Fn[k; ; ] (y) = 0;
n ¿ k + 1;
(75)
satis:ed by (Pn[k; ; ] )n are (with n ¿ k + 1)
; ]
(x) = [( − 1)k Pk −1 (x)Pk (x) + Pk2 (x)](x)Pn (x);
A[k;
n
Bn[k; ; ] (x) = [( − 1)k Pk −1 (x)Pk (x) + Pk2 (x)](x)Qn (x);
Cn[k; ; ] (x) = [0 k + ( − 1)k (x)Pk −1 (x)Qk (x) + (x)Pk (x)Qk (x)]Pn (x);
(76)
Dn[k; ; ] (x) = [0 k + ( − 1)k (x)Pk −1 (x)Qk (x) + (x)Pk (x)Qk (x)]Qn (x):
The co-dilated Pn[k; ; ] is related to these solutions by
( − 1)k (x)Pk −1 (x)Qk (x) + (x)Pk (x)Qk (x)
[k; ; ]
Pn (x)
= 1+
Pn
0 k
−
( − 1)k (x)Pk −1 (x)Pk (x) + (x)Pk2 (x)
Qn (x);
0 k
n ¿ k + 1:
(77)
316
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
5. Applications
5.1. Asymptotic formulas for the four solutions
We use results given in Theorem 2 and the asymptotic formula for the function of the second
kind (see [28, p. 98])
n
1
i=0 i
Qn (x) = −
1+O
n+1
(x)x
x
to get the following formulas for the solutions given in relation (59):
Theorem 7.
An(r) (x)
Bn(r) (x)
=x
n+2r −1
=−
n+r
i=0 i
xn+2
Cn(r) (x) = −xn
r −1
i=0
Dn(r) (x)
=
1
;
(x) 1 + O
x
i
1
;
1+O
x
1
;
1+O
x
(78)
r −1
i n+r
1
i=0 i
:
1+O
(x)xn+2r+1
x
i=0
5.2. Hypergeometric representation of the solutions
We give for each classical situation a hypergeometric representation of the polynomials Pn , the
function of the second kind Qn , four linearly independent solutions of the di&erential equation for
Pn(r) and relations between these solutions and the associated polynomials. In what follows, (a)k and
p Fq denote the Pochhammer symbol and the generalized hypergeometric function, respectively, and
are de2ned by
(a)k = a(a + 1) · · · (a + k − 1);
p Fq
k ∈ N; (a)0 ≡ 1;
∞
a1 ; a2 ; : : : ; ap (a1 )k (a2 )k · · · (ap )k xk
;
x =
(b1 )k (b2 )k · · · (bq )k k!
b1 ; b2 ; : : : ; bq k=0
where p and q belong to N, and x; a; ai and bi are complex numbers. The p Fq (x) is well de2ned
if no bi ; 1 6 i 6 q is a negative integer or zero and it constitutes a convergent series for all x if
p 6 q, or if p = q + 1 and |x| ¡ 1.
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
317
5.2.1. Monic Jacobi polynomials pn(+; )
We denote by pn(+; ) the monic Jacobi polynomials and qn(+; ) the corresponding function of the
second kind. The data are [28, p. 286]:
(x) = 1 − x2 ;
(x) = −(+ + + 2)x + − +;
(x) = (1 − x)+ (1 + x)
;
+ ¿ − 1; ¿ − 1;
I = [ − 1; 1];
n = n(n + + + + 1);
n =
2 − +2
;
(2n + + + )(2n + + + + 2)
n =
4n(n + +)(n + )(n + + + )
;
(2n + + + + 1)(2n + + + )2 (2n + + + − 1)
pn(+; )
2n (+ + 1)n
=
2 F1
(n + + + + 1)n
qn(+; ) =
−n; n + + + + 1 1 − x
;
2
++1
(−1)n 22n+++
+1 n!(n + + + 1)(n + + 1)
(1 − x)n+++1 (1 + x)
(n + + + + 1)n (2n + + + + 2)
× 2 F1
n + 1; n + + + 1 2
:
2n + + + + 2 1 − x
It should be mentioned that for n = 0 and + + + 1 = 0; q0(+; −1−+) is constant with respect to x.
In this case, the nonconstant solution of Eq. (5) is given in [34, p. 75].
From Theorem 2 and the previous data, four linearly independent solutions of the fourth-order
di&erential equation
(1 − x2 )2 y (x) − 10x(1 − x2 )y (x) + (−8 + 2x
2 − 2x+2 − +2 + 2n2 − 2n+x2 − 2n
x2
− 2x2 +
− 4nrx2 − 4r+x2 − 4r
x2 − 2nx2 − x2 +2 − 4r 2 x2 − 2n2 x2
− x2 2 + 2n + 4
r + 4+r + 24x2 + 2n+ + 4nr + 2n
+ 2+
− 2 + 4r 2 )y (x)
+ (−12xr 2 − 3x
2 − 12xnr − 6xn2 − 6xn − 3x+2 − 12xr
− 6x+
− 6xn
− 6xn+ − 12xr+ + 12x − 3+2 + 3
2 )y (x)
+ n(2 + n)(n + 1 + + + + 2r)(n − 1 + + + + 2r)y(x) = 0
(79)
318
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
satis2ed by the rth associated Jacobi orthogonal polynomials [36] (see also [5]) are:
1−x
1
−
r;
r
+
+
+
(r)
+
An; J = (1 − x) (1 + x) 2 F1
2
++1
−n − r; n + r + + + + 1 1 − x
;
×2 F1
2
++1
1−x
2
1
−
r;
r
+
+
+
n
+
r
+
1;
n
+
r
+
+
+
1
Bn;(r)J = (1 − x)−n−r −1 2 F1
;
2 F1
2
++1
2n + 2r + + + + 2 1 − x
r; r + + 2
−n − r; n + r + + + + 1 1 − x
(r)
−r
Cn; J = (1 − x) 2 F1
;
2 F1
2
2r + + + 1 − x
++1
2
−n−2r −+−1
r;
r
+
+
(1
−
x)
Dn;(r)J =
2 F1
(1 + x)
2r + + + 1 − x
n + r + 1; n + r + + + 1 2
:
(80)
×2 F1
2n + 2r + + + + 2 1 − x
The use of (60), the previous data and the fact that
r −1
i =
i=0
22r+++
−1 (2r + + + − 1)(r)(r + +)(r + )(r + + + )
;
(2r + + + )2
r ∈ N¿0 ;
allow us to represent the associated Jacobi polynomials Pn(r) (r ∈ N) in terms of the hypergeometric
functions (see also [36]).
Pn(r) (x)
(+; )
(+; )
(+; )
(1 − x)+ (1 + x)
(pr(+;−
)
1 (x)qn+r (x) − qr −1 (x)pn+r (x))
=
;
r −1
i=0 i
r ∈ N¿0 :
Remark 3. Note that An;(r)J ; Bn;(r)J ; Cn;(r)J and Dn;(r)J are multiples of the functions (59) in order to be as
simple as possible. This applies also for the Laguerre and Hermite case below.
Since 2 F1 constitutes a convergent series for |x| ¡ 1, functional relation (see [28, p. 270])
1
+; +;
1
+
+
−
()(
− +)
(−x)−+ 2 F1
x =
2 F1
(
)( − +)
1++−
x
1
;
1
+
−
()(+ − )
(−x)−
2 F1
(81)
+
(+)( − )
1+
−+ x
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
319
can be used in order to get for the functions qn+; ; An;(r)J ; Bn;(r)J ; Cn;(r)J and Dn;(r)J given above a representation with convergent series expansion when |x| ¿ 1.
5.2.2. Monic Laguerre polynomials l+n
We denote by l+n the monic Laguerre polynomials and qn+ the corresponding function of the second
kind. The data are [28, p. 286]:
(x) = x;
(x) = + + 1 − x;
(x) = x+ e−x ;
n = n;
+ ¿ − 1;
I = [0; ∞);
n = 2n + 1 + +;
l+n (x) = (−1)n (+ + 1)n
n = n(n + +);
−n x ;
1 F1
++1
qn+ (x) = (−1)n n!ei-+ (n + + + 1)ex G(n + + + 1; + + 1; −x);
where G(a; b; x) is the conRuent hypergeometric function of the second kind (see [28, p. 272]) and
related to the conRuent hypergeometric function 1 F1 by
+ + − + 1 (1 − )
( − 1) 1−
x 1 F1
G(+; ; x) =
(82)
x +
x :
1 F1
(+ − + 1)
(+)
2− In case of convergence, the function G(+; ; x) can also be represented by the hypergeometric function
2 F0 (see [1, Chapter 13]) as
+; 1 + + − 1
−+
:
(83)
G(+; ; x) = x 2 F0
−
x
−
From Theorem 2 and the previous data, four linearly independent solutions of the di&erential equation
[5] (see also [2])
x2 y (x) + 5xy (x) + (4 + 2xn + 4xr − x2 + 2x+ − +2 )y (x)
+ (6r − 3x + 3+ + 3n)y (x) + n(2 + n)y(x) = 0
satis2ed by the rth associated Laguerre orthogonal polynomials are:
1
−
r
−n
−
r
An;(r)L = x+ e−x 1 F1
x 1 F1
x ;
++1
++1 1 − r (r)
+
Bn; L = x 1 F1
x G(n + r + + + 1; + + 1; −x);
++1
(84)
320
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
Cn;(r)L
+
= x G(r + +; + + 1; −x)1 F1
−n − r x ;
++1 Dn;(r)L = x+ ex G(r + +; + + 1; −x)G(n + r + + + 1; + + 1; −x):
(85)
The use of (60), the previous data and the fact that
r −1
i = (r)(r + +);
i=0
allow us to represent the associated Laguerre polynomials Pn(r) (r ∈ N¿0 ) in terms of the hypergeometric functions
Pn(r) (x) =
(−1)n−1 ei-+ (n + r + + + 1)x+
(+ + 1)
1 − r (n + r + 1)
×
x G(n + r + + + 1; + + 1; −x)
1 F1
(r)
++1
−n − r − G(r + +; + + 1; −x)1 F1
; r ∈ N¿0 :
x
++1 5.2.3. Monic Hermite polynomials hn
We denote by hn the monic Hermite polynomials and by qn the corresponding function of the
second kind. The data are [28, p. 286]:
(x) = 1;
(x) = −2x;
2
(x) = e−x ;
I = (−∞; +∞);
n
n = 0; n = ;
2
n = 2n;
n 1
hn (x) = G − ; ; x2
2 2
qn (x) =
√
-n!2−n ex
2

n 1−n ;
−
1
= xn 2 F0  2 2 − 2  ;
x
−

+i-(n−1)=2
h−n−1 (ix)
√
n+1 1
−n x2 +i-(n−1)=2
2
; ; −x :
= -n!2 e
G
2 2
From Theorem 2 and the previous data, the four linearly independent solutions of the di&erential
equation [5] (see also [2])
y(iv) (x) + (4n + 8r − 4x2 )y (x) − 12xy (x) + 4n(2 + n)y(x) = 0
(86)
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
satis2ed by the rth associated Hermite orthogonal polynomials are:
n+r 1 2
1−r 1 2
(r)
−x 2
; ;x G −
; ;x ;
An; H = e G
2 2
2 2
n+r+1 1
1−r 1 2
; ;x G
; ; −x2 ;
Bn;(r)H = G
2 2
2
2
n+r 1 2
r 1
(r)
2
; ; −x G −
; ;x ;
Cn; H = G
2 2
2 2
n+r+1 1
r 1
2
; ; −x2 G
; ; −x2 :
Dn;(r)H = ex G
2 2
2
2
321
(87)
By using (60), the previous data and the fact that
r −1
i =
√
-(r)21−r ;
i=0
we represent the associated Hermite orthogonal polynomials Pn(r) (r ∈ N¿0 ) as
1−r 1 2
n+r+1 1
(r)
i-(r −2)=2
−n−1 i-(n+1)=2 (n + r)
2
G
; ;x G
; ; −x
Pn (x) = e
2
e
(r)
2 2
2
2
n+r 1 2
r 1
; ; −x2 G −
; ;x
; r ∈ N¿0 :
− G
2 2
2 2
5.3. Extension of results to real order of association
Let 3 be a real number with 3 ¿ 0 and (Pn(3) )n the family of polynomials de2ned by
(3)
(x) = (x − n+3 )Pn(3) (x) − n+3 Pn(3)
Pn+1
−1 (x);
n¿1
(88)
with the initial conditions
P0(3) (x) = 1;
P1(3) (x) = x − 3 ;
(89)
where n+3 and n+3 are the coe@cients n and n of Eq. (1) with n replaced by n + 3.
We assume that the starting family (Pn )n de2ned in (1) is classical. The coe@cients n and n
are therefore rational functions in the variable n [18,19,28] and the coe@cients n+3 and n+3 are
well de2ned. When n+3 = 0; ∀n ¿ 1, the family (Pn(r) )n , thanks to Favard’s theorem [12,8], is
orthogonal and represents the associated of the family (Pn )n with real order of association. The
notion of associated orthogonal polynomials with real order of association has been investigated by
several authors (see for example [2,6,20,36]).
322
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
Theorem 8. Let (Pn )n be a family of classical orthogonal polynomial, 3 ¿ 0 a real number and
(Pn(3) )n the 3-associated of (Pn )n . We have:
(1) (Pn(3) )n satis:es
Fn(3) (y) = 0;
(90)
where Fn(3) is the operator given in (51) with r replaced by 3.
(2) The di>erential operator Fn(3) factorizes as
Sn(3) Tn(3) = U33−1 Fn(3) ;
S̃n(3) T̃n(3) = V33−1 Fn(3) ;
(91)
where the operators Sn(3) ; Tn(3) ; S̃n(3) ; T̃n(3) are those given in Eqs. (48)–(55) with r replaced by
3; Pr and Qr are replaced by U3 and V3 , respectively. U3 and V3 are the two linearly independent
solutions of the di>erential equation (see [28])
(x)y (x) + (x)y (x) + 3 y(x) = 0
with Ur = Pr ; Vr = Qr for 3 = r ∈ N and
3
3 = − ((3 − 1) + 2 ):
2
Four linearly independent solutions of the di>erential equation (90) are given by
(92)
(93)
An(3) (x) = (x)U3−1 (x)Un+3 (x);
Bn(3) (x) = (x)U3−1 (x)Vn+3 (x);
Cn(3) (x) = (x)V3−1 (x)Un+3 (x);
(94)
Dn(3) (x) = (x)V3−1 (x)Vn+3 (x);
where (x) is the weight function given by (3).
Proof. (1) Let n be a 2xed integer number and de2ne the function 6 by
6:
R+ → R;
3 → Fn(3) (Pn(3) (x));
where R+ is the set of positive real numbers. Using relation (88) for 2xed x; 6(3) can be written
as rational function in 3. In fact, for the classical orthogonal polynomials, the three-term recurrence
relation coe@cients n and n are rational functions in the variable n. Using Eq. (58) we get
6(r) = Fn(r) (Pn(r) (x)) = 0;
∀r ∈ N:
(95)
We then conclude that 6(3) is a rational function with an in2nite number of zeros. Therefore,
6(3) = 0; ∀3 ∈ R+ , and (Pn(3) )n satis2es (90).
(2) Eq. (91) are proved by a straightforward computation using that U3 and U3 satisfy (92).
(3) The functions given in (94) are represented as products of functions satisfying homogeneous
di&erential equation of order 1 (for ) and 2 (for U and V ). These functions therefore satisfy
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
323
a di&erential equation of order 4(=1 × 2 × 2) which is identical to (90). Notice that by linear algebra
one can deduce the di&erential equation of the product (94), given the di&erential equations of the
factors, since they have polynomial coe@cients. This can be done, e.g., by the Maple command
‘diffeq*diffeq’ [33] 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 orthogonal polynomials with real
order of association as was done for the classical discrete case in [13].
5.4. Solutions of some second-order di>erential equations
Proposition 3. The two linearly independent solutions of the di>erential equation
S(r) (y) = 0;
where the operator S(r) (see (48)) is given by
Sn(r) = Pr −1 D2 + [( + )Pr −1 − 2Pr −1 ]D + [( + n+r − r −1 )Pr −1 − 2Pr −1 ]
are:
(x));
En(r) (x) = (x)(x)(Qr −1 (x)Pr −1 (x) − Pr −1 (x)Qr −1 (x))(Pn+r (x)Pr −1 (x) − Pr −1 (x)Pn+r
(x)):
Fn(r) (x) = (x)(x)(Qr −1 (x)Pr −1 (x) − Pr −1 (x)Qr −1 (x))(Qn+r (x)Pr −1 (x) − Pr −1 (x)Qn+r
Proposition 4. The two linearly independent solutions of the di>erential equation
S(3) (y) = 0;
where the operator S(3) (see (94)) is given by
S̃n(3) = U3−1 D2 + [( + )U3−1 − 2U3−1 ]D + [( + n+3 − 3−1 )U3−1 − 2U3−1 ]
are:
En(3) (x) = (x)(x)(V3−1 (x)U3−1 (x) − U3−1 (x)V3−1 (x))(Un+3 (x)U3−1 (x) − U3−1 (x)Un+3
(x));
(x)):
Fn(3) (x) = (x)(x)(V3−1 (x)U3−1 (x) − U3−1 (x)V3−1 (x))(Vn+3 (x)U3−1 (x) − U3−1 (x)Vn+3
Here, U3 and V3 are solutions of (92).
Proof. Since the functions Cn(r) and Dn(r) are solutions of equation Fn(r) (y) = 0 (see Theorem 2), we
use the factorization given by (50) and get
Sn(r) (Tn(r) (y)) = Pr3−1 Fn(r) (y) = 0
for y ∈ {Cn(r) ; Dn(r) }. We therefore conclude that the functions En(r) and Fn(r) de2ned by
En(r) = Tn(r) (Cn(r) );
Fn(r) = Tn(r) (Dn(r) );
satisfy Sn(r) (y) = 0. Computations using the fact that Pn and Qn satisfy (5) lead to the expressions
given in Proposition 3. The proof of Proposition 4 is similar to the one of Proposition 3 by using
Theorem 7.
324
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
Remark 4. (1) The previous propositions give solutions to families of second-order di&erential equations. In particular, Proposition 3 solves a family of second-order di&erential equations with polynomial coe@cients.
(2) For 2xed integer r, and for di&erent classical situations, we have tried without success to
solve the di&erential equation Sn(r) (y) = 0 using Maple 7 or Mathematica 4.1 [37].
(3) Mark van Hoeij [35] was able to solve di&erential equation Sn(r) (y) = 0 for 2xed integers
r using an algorithm he is currently developing and which extends the capabilities of algorithms
aimed at solving second-order linear homogeneous di&erential equation with polynomial
coe@cients.
5.5. Extension of results to semi-classical cases
The proof of Theorem 1, which is the starting point of this paper, uses merely the second-order
di&erential equation (5) and relation (33). Now, we suppose that the family (Pn )n is semi-classical
[3,17,25,26]. This implies that (Pn )n is orthogonal satisfying a second-order di&erential equation of
the form
O n (y(x)) = I2 (x; n)y (x) + I1 (x; n)y (x) + I0 (x; n)y(x) = 0;
M
(96)
where the coe@cients Ii (x; n) are polynomials in x of degree not depending on n. For semi-classical
orthogonal polynomials an equation of type (33) is known and can be stated as [4,5]
M̃n (Pn(1)
−1 (x)) = a1 (x)Pn (x) + a0 (x)Pn (x);
(97)
where ai are polynomials and M̃n a second-order linear di&erential operator with polynomial coe@cients. Use of the two previous equations leads to the following extension.
Theorem 9. Given (Pn )n a sequence of semi-classical orthogonal polynomials satisfying (96) and
(PO n )n a family of orthogonal polynomials obtained by modifying (Pn )n and satisfying
(1)
PO n (x) = An (x)Pn+k
−1 + Bn (x)Pn+k ;
n ¿ k ;
(98)
where An and Bn are polynomials of degree not depending on n, and k; k ∈ N, we have the following:
(1) The orthogonal polynomials (PO n )n¿k satisfy a common fourth-order linear di>erential equation
FOn (y(x)) = K4 (x; n)y (x) + K3 (x; n)y (x) + K2 (x; n)y (x)
+ K1 (x; n)y (x) + K0 (x; n)y(x) = 0;
(99)
where the coe7cients Ki are polynomials in x, with degree not depending on n.
(2) The operator FOn can be factored as product of two second-order linear di>erential operators
XO n FOn = SO n TO n ;
where XO n and the coe7cients of SO n and TO n are polynomials of degree not depending on n.
(100)
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
325
The proof is similar to the one of Theorem 1 but with (96) and (97) playing the role of (5) and
(33), respectively.
The previous theorem covers the modi2cations such as the rth associated, the general co-recursive,
the general co-dilated, the general co-recursive associated and the general co-modi2ed semi-classical
orthogonal polynomials.
When the orthogonal polynomial sequence (Pn )n is semi-classical, in general it is di@cult to
O n , M̃n ; FOn ; SO n and TO n in terms of the
represent the coe@cients of the di&erential operators M
polynomials and , the coe@cients of the functional equation (see [26, p. 37]) satis2ed by the
regular functional with respect to which (PO n )n is orthogonal.
However, for particular cases (for example if the degrees of polynomials and are small), it is
O n , M̃n ; FOn ; SO n
possible after huge computations to give the coe@cients of the di&erential operators M
and TO n explicitly, and therefore look for functions annihilating these di&erential operators.
Acknowledgements
We are very grateful to the Alexander von Humboldt Foundation whose grant (Fellowship No IV
KAM 1070235) accorded to the 2rst author (M.F.) was decisive for the realization of this work.
Also, A.R. and M.F. acknowledge the kind hospitality of the Computational Mathematics group of
the University of Kassel as well as Professor Anne LemaT̂tre (UnitUe des SystVemes Dynamiques)
of the FacultUes Universitaires Notre-Dame de la Paix, during their visits to Kassel and Namur,
respectively.
References
[1] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions, National Bureau of Standards, Applied
Mathematics Series, Vol. 55, US Government, Washington, DC, 1964.
[2] R. Askey, J. Wimp, Associated Laguerre and Hermite polynomials, Proc. Roy. Soc. Edinburgh 96A (1984) 15–37.
[3] S. Belmehdi, On semi-classical linear functionals of class s = 1 Classi2cation and integral representation, Idag. Math.
(N.S.) 3 (3) (1992) 253–276.
[4] S. Belmehdi, A. Ronveaux, Polynômes associUes des polynômes orthogonaux classiques. Construction via “REDUCE”,
in: L. Arias, et al. (Eds.), Orthogonal Polynomials and their Applications, Universidad de Oviedo, 1989, pp. 72–83.
[5] S. Belmehdi, A. Ronveaux, The fourth-order di&erential equation satis2ed by the associated orthogonal polynomials,
Rend. Mat. Ser. VII 11 (1991) 313–326.
[6] J. Bustoz, M.E.H. Ismail, The associated ultraspherical polynomials and their q-analogues, Canad. J. Math. XXXIV
(3) (1982) 718–736.
[7] T.S. Chihara, On co-recursive orthogonal polynomials, Proc. Amer. Math. Soc. 8 (1957) 899–905.
[8] T.S. Chihara, Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.
[9] J. Dini, Sur les formes linUeaires et polynômes orthogonaux de Laguerre–Hahn, ThVese de Doctorat, UniversitUe Pierre
et Marie Curie, Paris VI, 1988.
[10] J. Dini, P. Maroni, A. Ronveaux, Sur une perturbation de la rUecurrence vUeri2Uee par une suite de polynômes
orthogonaux, Portugal. Math. 6 (3) (1989) 269–282.
[11] J. Dzoumba, Sur les Polynômes de Laguerre–Hahn, ThVese de 3Veme cycle, UniversitUe Pierre et Marie Curie, Paris
VI, 1985.
[12] J. Favard, Sur les polynômes de Tchebiche&, C. R. Acad. Sci. Paris 200 (1935) 2052–2053.
[13] M. Foupouagnigni, W. Koepf, A. Ronveaux, On fourth-order di&erence equations for orthogonal polynomials of a
discrete variable: derivation, factorization and solutions, J. Di&erence Equations Appl. 9 (2003) 777–804.
326
M. Foupouagnigni et al. / Journal of Computational and Applied Mathematics 162 (2004) 299 – 326
[14] C.C. Grosjean, Theory of recursive generation of systems of orthogonal polynomials: an illustrative example, J.
Comput. Appl. Math. 12& 13 (1985) 299–318.
[15] C.C. Grosjean, The weight function, generating functions and miscellaneous properties of the sequences of orthogonal
polynomials of the second kind associated with the Jacobi and the Gegenbauer polynomials, J. Comput. Appl. Math.
16 (1986) 259–307.
[16] W. Hahn, On di&erential equations for orthogonal polynomials, Funkcial. Ekvac. 21 (1978) 1–9.
[17] E. Hendriksen, H. van Rossum, Semi-classical orthogonal polynomials, in: C. Brezinski et al. (Eds.), Polynômes
Orthogonaux et Applications, Lecture Notes in Mathematics, Vol. 1171, Springer, Berlin, 1985, pp. 355 –361.
[18] W. Koepf, D. Schmersau, Representations of orthogonal polynomials, J. Comput. Appl. Math. 90 (1998) 57–94.
[
[19] P. Lesky, Uber
Polynoml[osungen von Di&erentialgleichungen und Di&erenzengleichungen zweiter Ordnung, Anz.
[
Osterreich.
Akad. Wiss. Math.-Natur. Kl. 121 (1985) 29–33.
[20] J. Letessier, Some results on co-recursive associated Laguerre and Jacobi polynomials, SIAM J. Math. Anal. 25 (2)
(1994) 528–548.
[21] S. Lewanowicz, Results on the associated classical orthogonal polynomials, J. Comput. Appl. Math. 65 (1995)
215–231.
[22] A. Magnus, Riccati acceleration of Jacobi continued fractions and Laguerre–Hahn orthogonal polynomials, Lecture
Notes in Mathematics, Vol. 1071, Springer, Berlin, 1984, pp. 213–230.
[23] F. MarcellUan, J.S. Dehesa, A. Ronveaux, On orthogonal polynomials with perturbed recurrence relations, J. Comput.
Appl. Math. 30 (2) (1990) 203–212.
[24] F. MarcellUan, E. Prianes, Perturbations of Laguerre–Hahn functionals, J. Comput. Appl. Math. 105 (1999) 109–128.
[25] P. Maroni, ProlUegomVenes aV l’Uetude des polynômes orthogonaux semi-classiques, Ann. Mat. Pura Appl. 4 (1987)
165–184.
[26] P. Maroni, Une thUeorie algUebrique des polynômes orthogonaux: Applications aux polynômes orthogonaux
semi-classiques, in: C. Brezinski, et al., (Eds.), Orthogonal Polynomials and Applications, Annals on Computing
and Applications of Mathematics, Vol. 9, J.C. Baltzer AG, Basel, 1991, pp. 98–130.
[27] M.B. Monagan, K.O. Geddes, K.M. Heal, G. Labahn, S.M. Vorkoetter, J. McCarron, P. DeMarco, Maple 7, Waterloo
Maple, Inc., 2001.
[28] A.F. Nikiforov, V.B. Uvarov, Special Functions of Mathematical Physics, Birkh[auser, Basel, Boston, 1988.
[29] A. Ronveaux, Fourth-order di&erential equation for numerator polynomials, J. Phys. A.: Math. Gen. 21 (1988)
749–753.
[30] A. Ronveaux, S. Belmehdi, J. Dini, P. Maroni, Fourth-order di&erential equation for the co-modi2ed semi-classical
orthogonal polynomials, J. Comput. Appl. Math. 29 (2) (1990) 225–231.
[31] A. Ronveaux, F. MarcellUan, Co-recursive orthogonal polynomials and fourth-order di&erential equation, J. Comput.
Appl. Math. 25 (1989) 295–328.
[32] A. Ronveaux, A. Zarzo, E. Godoy, Fourth-order di&erential equation satis2ed by the generalized co-recursive of all
classical orthogonal polynomials. A study of their distribution of zeros, J. Comput. Appl. Math. 59 (1995) 105–109.
[33] B. Salvy, P. Zimmermann, GFUN: A Maple package for the manipulation of generating and holonomic functions in
one variable, ACM Trans. Math. Software 20 (1994) 163–177.
[34] G. Szeg[o, Orthogonal Polynomials, Vol. XXIII, American Mathematical Society, Colloquium Publications, 1939.
[35] M. van Hoeij, Private communication, May 15, 2002.
[36] J. Wimp, Explicit formulas for the associated Jacobi polynomials and applications, Canad. J. Math. XXXIX (4)
(1987) 983–1000.
[37] S. Wolfram, The Mathematica Book, 4th Edition, Wolfram Media, Cambridge University Press, Cambridge 1999.
[38] A. Zarzo, A. Ronveaux, E. Godoy, Fourth-order di&erential equation satis2ed by the associated of any order of all
classical orthogonal polynomials. A study of their distribution of zeros, J. Comput. Appl. Math. 49 (1993) 349–359.