Journal article
On solutions of holonomic divideddifference equations on nonuniform lattices
Publication Details
Authors:  Foupouagnigni, M.; Koepf, W.; Kenfack Nangho, M.; Mboutngam, S.

Abstract
The main aim of this paper is the development of suitable bases that enable the direct series representation of orthogonal polynomial systems on nonuniform lattices (quadratic lattices of a discrete or a qdiscrete variable). We present two bases of this type, the first of which allows one to write solutions of arbitrary divideddifference equations in terms of series representations, extending results given by Sprenger for the qcase. Furthermore, it enables the representation of the Stieltjes function, which has already been used to prove the equivalence between the Pearson equation for a given linear functional and the Riccati equation for the formal Stieltjes function. If the AskeyWilson polynomials are written in terms of this basis, however, the coefficients turn out to be not qhypergeometric. Therefore, we present a second basis, which shares several relevant properties with the first one. This basis enables one to generate the defining representation of the AskeyWilson polynomials directly from their divideddifference equation. For this purpose, the divideddifference equation must be rewritten in terms of suitable divideddifference operators developed in previous work by the first author.