Recurrence equations and their classical orthogonal polynomial solutions

The classical orthogonal polynomials are given as the polynomial solutions pnðxÞ of the differential equation rðxÞy00ðxÞ þ sðxÞy0ðxÞ þ knyðxÞ 1⁄4 0; 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 pnþ1ðxÞ 1⁄4 ðAnxþ BnÞpnðxÞ Cnpn 1ðxÞ 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. In a similar way, classical discrete orthogonal polynomial solutions of holonomic three-term recurrence equations can be determined by considering their corresponding difference equation rðxÞDryðxÞ þ sðxÞDyðxÞ þ knyðxÞ 1⁄4 0; where DyðxÞ 1⁄4 yðxþ 1Þ yðxÞ and ryðxÞ 1⁄4 yðxÞ yðx 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 Applied Mathematics and Computation 128 (2002) 303–327 www.elsevier.com/locate/amc Corresponding author. Present address: Department of Mathematics, University of Kassel, Heinrich-Plett-Str. 40, D-34132 Kassel, Germany. E-mail address: [email protected] (W. Koepf). 0096-3003/02/$ see front matter 2002 Elsevier Science Inc. All rights reserved. PII: S0096 -3003 (01 )00078-9 rðxÞDqD1=qyðxÞ þ sðxÞDqyðxÞ þ kq;nyðxÞ 1⁄4 0; where Dqf ðxÞ 1⁄4 f ðqxÞ f ðxÞ ðq 1Þx ; q 61⁄4 1; denotes the q-difference operator. 2002 Elsevier Science Inc. All rights reserved.