Limits of elliptic hypergeometric biorthogonal functions

Abstract

The purpose of this article is to bring structure to (basic) hypergeometric biorthogonal systems, in particular to the -Askey scheme of basic hypergeometric orthogonal polynomials. We aim to achieve this by looking at the limits as p -> 0 of the elliptic hypergeometric biorthogonal functions from Spiridonov (2003), with parameters which depend in varying ways on p. As a result we get 38 systems of biorthogonal functions with for each system at least one explicit measure for the bilinear form. Amongst these we indeed recover the q-Askey scheme. Each system consists of (basic hypergeometric) rational functions or polynomials.