Hypergeometric Solutions of the $A_4^{(1)}$-Surface $q$-Painlev\'e IV Equation

TL;DR

This paper explores the $A_4^{(1)}$-surface $q$-Painlevé IV equation, presenting three types of classical solutions with determinantal structures involving basic hypergeometric functions, including ${}_2 heta_1$ and ${}_2 heta_2$ series.

Contribution

It introduces three new classical solutions to the $A_4^{(1)}$-surface $q$-Painlevé IV equation, characterized by determinantal formulas with hypergeometric series.

Findings

Three types of classical solutions with determinantal structures

Solutions expressed via ${}_2 heta_1$ and ${}_2 heta_2$ hypergeometric series

Explicit connection to hypergeometric functions in integrable systems

Abstract

We consider a $q$-Painlev\'e IV equation which is the $A_4^{(1)}$-surface type in the Sakai's classification. We find three distinct types of classical solutions with determinantal structures whose elements are basic hypergeometric functions. Two of them are expressed by ${}_2\varphi_1$ basic hypergeometric series and the other is given by ${}_2\psi_2$ bilateral basic hypergeometric series.