Recurrence relations for the connection coefficients of classical orthogonal polynomials. Issue 7 (3rd July 2018) - Mu, Yan-Ping
ABSTRACT: In this paper, we present a framework on deriving recurrence relations for the connection coefficients of orthogonal polynomials. Let{ P n ( x ) } n ≥ 0 and{ Q n ( x ) } n ≥ 0 be two sequences of classical orthogonal polynomials. Assume that there exists a non-trivial linear operator L such that{ L P n ( x ) } n ≥ 0 and{ L Q n ( x ) } n ≥ 0 are also orthogonal polynomials. We show that the connection coefficients between{ P n ( x ) } n ≥ 0 and{ Q n ( x ) } n ≥ 0 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.