Non-polynomial $q$-Askey scheme: integral representations, eigenfunction properties, and polynomial limits

TL;DR

This paper introduces a non-polynomial extension of the $q$-Askey scheme using contour integrals with special functions, revealing new eigenfunction properties and polynomial limits.

Contribution

It constructs a novel non-polynomial $q$-Askey scheme based on integral representations involving hyperbolic gamma functions and related special functions.

Findings

Elements are expressed as contour integrals with hyperbolic gamma functions.

Elements exhibit joint eigenfunction properties.

The scheme includes polynomial limits of the non-polynomial elements.

Abstract

We construct a non-polynomial generalization of the $q$-Askey scheme. Whereas the elements of the $q$-Askey scheme are given by $q$-hypergeometric series, the elements of the non-polynomial scheme are given by contour integrals, whose integrands are built from Ruijsenaars' hyperbolic gamma function. Alternatively, the integrands can be expressed in terms of Faddeev's quantum dilogarithm, Woronowicz's quantum exponential, or Kurokawa's double sine function. We present the basic properties of all the elements of the scheme, including their integral representations, joint eigenfunction properties, and polynomial limits.