q-Heun equation and initial-value space of q-Painlev\'e equation

TL;DR

This paper explores the connection between the q-Heun equation and q-Painlevé equations, revealing how certain linear q-difference equations emerge from their initial-value spaces and deriving a specific operator from these relations.

Contribution

It demonstrates the appearance of the q-Heun equation in the context of q-Painlevé equations and derives the degenerated Ruijsenaars-van Diejen operator from these structures.

Findings

q-Heun equation appears in linear q-difference equations of q-Painlevé

Initial-value space blow-up relates to q-Heun and q-Painlevé equations

Derived the degenerated Ruijsenaars-van Diejen operator from q-Painlevé context

Abstract

We show that the q-Heun equation and its variants appear in the linear q-difference equations associated to some q-Painlev\'e equations by considering the blow-up associated to their initial-value spaces. We obtain the firstly degenerated Ruijsenaars-van Diejen operator from the linear q-difference equation associated to the q-Painlev\'e equation of type $E_8$.