Hypergeometric solutions to the q-Painlev\'e equations

TL;DR

This paper constructs hypergeometric solutions for seven q-Painlevé equations using geometric methods to reduce these equations to recurrence relations for q-hypergeometric functions.

Contribution

It introduces a geometric approach to derive explicit hypergeometric solutions for q-Painlevé equations, expanding the solution space for these nonlinear difference equations.

Findings

Explicit hypergeometric solutions for seven q-Painlevé equations

Reduction of q-Painlevé equations to three-term recurrence relations

Application of plane curve geometry in solving nonlinear difference equations

Abstract

Hypergeometric solutions to seven q-Painlev\'e equations in Sakai's classification are constructed. Geometry of plane curves is used to reduce the q-Painlev\'e equations to the three-term recurrence relations for q-hypergeometric functions.