q-Special functions, an overview

TL;DR

This overview introduces $q$-special functions, covering their theory, recent developments, and applications in areas like quantum groups, combinatorics, and statistical mechanics, emphasizing $q$-hypergeometric series and orthogonal polynomials.

Contribution

It provides a comprehensive summary of $q$-special functions, including recent topics like nonsymmetric analogues and $q=-1$ limits, and discusses their applications across various mathematical and physical fields.

Findings

Overview of $q$-hypergeometric series and identities

Discussion of $q$-orthogonal polynomials, especially Askey--Wilson polynomials

Connections to quantum groups, Lie algebras, and statistical mechanics

Abstract

This article gives a brief introduction to $q$-special functions, i.e., $q$-analogues of the classical special functions. Here $q$ is a deformation parameter, usually $0<q<1$, where $q=1$ is the classical case. The main topics to be treated are $q$-hypergeometric series, with some selected evaluation and transformation formulas, and the $q$-hypergeometric orthogonal polynomials, most notably the Askey--Wilson polynomials. Some newer topics as nonsymmetric analogues and $q=-1$ limits will also be addressed. In several variables we discuss Macdonald polynomials associated with root systems, in particular the $A_n$ and the $BC_n$ case. The theory of elliptic hypergeometric series also gets some attention. The occurrence of $q$-series in number theory and combinatorics will be discussed. Finally we indicate applications and interpretations in quantum groups, Chevalley groups, affine Lie algebras and statistical mechanics.