Recurrence equations and their classical orthogonal polynomial solutions

The classical orthogonal polynomials are given as the polynomial solutions p, (x) of the differential equation sigma(x)y"(x) + tau(x)y'(x) + lambda(n)y(x) = 0, where sigma(x) is a polynomial of at most second degree and tau(x) is a polynomial of first degree. In this paper a general method to express the coefficients A(n), B-n and C-n of the recurrence equation p(n+1)(x) = (A(n)x + B-n)p(n)(x) - C(n)p(n-1)(x) in terms of the given polynomials sigma(x) and tau(x) is used to present an algorithm to determine the classical orthogonal polynomial solutions of any given holonomic three-term 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 sigma(x)Deltadely(x) + tau(x)Deltay(x) + lambda(n)y(x) = 0, where Deltay(x) = y(x + 1) - y(x) and Deltay(x) = y(x) - y(x - 1) 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 sigma(x)D(q)D(1/q)y(x) + tau(x)D(q)y(x) + lambda(q,n)y(x) = 0. where D(q)f(x) = f(qx) - f(x)/(q-1)x, qnot equal 1. denotes the q-difference operator. (C) 2002 Elsevier Science Inc. All rights reserved.