Recurrence equations and their classical orthogonal

where rðxÞ is a polynomial of at most second degree and sðxÞ is a polynomial of first degree.
In this paper a general method to express the coefficients An; Bn and Cn of the recurrence equation
rðxÞDqD1=qyðxÞ þ sðxÞDqyðxÞ þ kq;nyðxÞ ¼ 0;
ð6Þ
where
f ðqxÞ À f ðxÞ Dqf ðxÞ ¼ ðq À 1Þx ; q 6¼ 1;
denotes the q-difference operator [6], and rðxÞ ¼ ax2 þ bx þ c and sðxÞ ¼ dx þ e are again polynomials of at most second- and of first-order, respectively. By equating the coefficients of xn in (6) one gets
Applied Mathematics and Computation 128 (2002) 303–327 /locate/amc
Recurrence equations and their classical orthogonal polynomial solutions
kn ¼ Àðanðn À 1Þ þ dnÞ:
ð4Þ
Similarly, a family pnðxÞ of polynomials of degree exactly n, given by (2), is a family of classical discrete orthogonal polynomials if it is the solution of a difference equation of the type
In a similar way, classical discrete orthogonal polynomial solutions of holonomic three-term recurrence equations can be determined by considering their corresponding difference equation
In this paper an algorithm is developed which answers this question for a large class of classical orthogonal polynomial systems. Furthermore, we present results of our corresponding Maple implementation retode and compare these with the Maple implementation rec2ortho of Koornwinder and Swarttouw [12]. These programs overlap, but rec2ortho does not cover Bessel, Hahn and q-polynomials, whereas retode does not include the Meixner–Pollaczek case.
kq;n ¼ Àa½n1=q½n À 1q À d½nq;
ð7Þ
where the abbreviation
½nq
¼
1 À qn 1Àq
denotes the so-called q-brackets. Note that limq!1½nq ¼ n. It can be shown (see e.g. [14]) that any solution pnðxÞ of either (3), (5) or (6)
rðxÞy00ðxÞ þ sðxÞy0ðxÞ þ knyðxÞ ¼ 0;
ð3Þ
W. Koepf, D. Schmersau / Appl. Math. Comput. 128 (2002) 303–327
305
where rðxÞ ¼ ax2 þ bx þ c is a polynomial of at most second-order and sðxÞ ¼ dx þ e ðd 6¼ 0Þ is a polynomial of first-order ([3,13]). Since one demands that pnðxÞ has exact degree n, by equating the coefficients of xn in (3) one gets
rðxÞDryðxÞ þ sðxÞDyðxÞ þ knyðxÞ ¼ 0;
ð5Þ
where
DyðxÞ ¼ yðx þ 1Þ À yðxÞ and ryðxÞ ¼ yðxÞ À yðx À 1Þ
denote the forward and backward difference operators, respectively, and rðxÞ ¼ ax2 þ bx þ c and sðxÞ ¼ dx þ e are again polynomials of at most secondand of first-order, respectively, see e.g. [18]. Again, (4) follows.
rðxÞDryðxÞ þ sðxÞDyðxÞ þ knyðxÞ ¼ 0; where DyðxÞ ¼ yðx þ 1Þ À yðxÞ and ryðxÞ ¼ yðxÞ À yðx À 1Þ denote the forward and backward difference operators, respectively, and a similar approach applies to classical q-orthogonal polynomials, being solutions of the q-difference equation
Keywords: Computer algebra; Maple; Differential equation; Q-difference equation; Structure formula
1. Introduction
Families of orthogonal polynomials pnðxÞ (corresponding to a positive-definite measure) satisfy a three-term recurrence equation of the form
0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 1 ) 0 0 0 7 8 - 9
304
W. Koepf, D. Schmersau / Appl. Math. Comput. 128 (2002) 303–327
Wolfram Koepf a,*, Dieter Schmersau b
a Dept. IMN, Hochschule f€ur Technik, Wirtschaft und Kultur Leipzig, D-04251 Leipzig, Germany b Freie Universit€at Berlin, Berlin, Germany
satisfies a recurrence equation (1).
The following is a general procedure to find the coefficients of the recurrence
* Corresponding author. Present address: Department of Mathematics, University of Kassel, Heinrich-Plett-Str. 40, D-34132 Kassel, Germany.
E-mail address: koepf@mathematik.uni-kassel.de (W. Koepf).
On the other hand, in practice one is often interested in an explicit solution of a given recurrence equation. Therefore it is an interesting question to ask whether a given recurrence equation has classical orthogonal polynomial solutions.
pnþ1ðxÞ ¼ ðAnx þ BnÞpnðxÞ À CnpnÀ1ðxÞ ðn 2 N0; pÀ1 0Þ
ð1Þ
with CnAnAnÀ1 > 0, see e.g. [5, p. 20]. Moreover, Favard’s theorem states that the converse is also true.
pnþ1ðxÞ ¼ ðAnx þ BnÞpnðxÞ À CnpnÀ1ðxÞ in terms of the given polynomials rðxÞ and sðxÞ is used to present an algorithm to determine the classical orthogonal polynomial solutions of any given holonomic threeterm recurrence equation, i.e., a homogeneous linear three-term recurrence equation with polynomial coefficients.