Some connection and linearization problems for polynomials in and beyond the Askey scheme

Abstract

The connection problem is considered in a hypergeometric function framework for (i) the two most general families of polynomials belonging to the Askey scheme (Wilson and Racah), and (ii) some generalized Laguerre and Jacobi polynomials falling outside that scheme (Sister Celine, Cohen and Prabhakar–Jain), which are relevant to the study of quantum-mechanical systems and include as particular cases, the generalizations of the classical families with Sobolev-type orthogonality. In addition, using the same method three new linearization-like formulae for the Gegenbauer polynomials are also derived: a linearization formula that generalizes the m=n case of Dougall's formula, the analogue of the m=n case of Nielsen's inverse linearization formula for Hermite polynomials, and a connection formula for the squares. Closed analytical formulae for the corresponding connection and linearization coefficients are given in terms of hypergeometric functions of unit argument, which at times can be further simplified and expressed as single hypergeometric terms.