Degenerations of $q$-Heun equation

TL;DR

This paper explores various degenerations of the $q$-Heun equation derived from $q$-Painlevé equations, defining new confluent forms and analyzing their limits to differential equations.

Contribution

It introduces definitions for confluent, biconfluent, and doubly confluent $q$-Heun equations and studies their limit procedures to classical differential equations.

Findings

Defined new confluent $q$-Heun equations

Established limit procedures to differential equations

Connected $q$-difference equations with classical limits

Abstract

We obtain several degenerations of the $q$-Heun equation by considering the linear $q$-difference equations associated to several $q$-Painlev\'e equations. We establish definitions of the confluent $q$-Heun equation, the biconfluent $q$-Heun equation and the doubly confluent $q$-Heun equation, and investigate limit procedures to the corresponding differential equations.