Hypergeometric Solutions for the $q$-Painlev\'e Equation of type $E^{(1)}_6$ by Pad\'e Method

TL;DR

This paper derives solutions for the $q$-Painlevé $E^{(1)}_6$ equation using Padé approximation, linking it to Bäcklund transformations and providing determinant formulas for special solutions.

Contribution

It introduces a novel application of the Padé method to the $q$-Painlevé $E^{(1)}_6$ equation and establishes its connection with Bäcklund transformations.

Findings

Derived explicit solutions using Padé method

Established relation between Padé method and Bäcklund transformations

Presented determinant formulas for special solutions

Abstract

The $q$-Painlev\'e equation of type $E^{(1)}_6$ is obtained by Pad\'e method. Special solutions in determinant formula to the $q$-Painlev\'e equation is presented. A relation between Pad\'e method and B\"acklund transformation of type $E^{(1)}_6$ is given.