Limits of elliptic hypergeometric biorthogonal functions

TL;DR

This paper explores the limits of elliptic hypergeometric biorthogonal functions as a parameter approaches zero, revealing 38 systems that include the q-Askey scheme and providing explicit measures for their bilinear forms.

Contribution

It systematically analyzes the p->0 limits of elliptic hypergeometric biorthogonal functions, establishing a structured classification including the q-Askey scheme.

Findings

Recovered the q-Askey scheme within the limit systems

Identified 38 biorthogonal systems with explicit measures

Included rational functions and polynomials in the systems

Abstract

The purpose of this article is to bring structure to (basic) hypergeometric biorthogonal systems, in particular to the q-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, 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.