Recurrence relations for the connection coefficients of classical orthogonal polynomials

Abstract

ABSTRACTIn this paper, we present a framework on deriving recurrence relations for the connection coefficients of orthogonal polynomials. Let and be two sequences of classical orthogonal polynomials. Assume that there exists a non-trivial linear operator L such that and are also orthogonal polynomials. We show that the connection coefficients between and satisfy a linear recurrence of order at most 4. Based on the hypergeometric representation of orthogonal polynomials, this recurrence relation can be computed by the aid of the extended Zeilberger algorithm. We illustrate our method by several examples.