The Pad\'e interpolation method applied to $q$-Painlev\'e equations

TL;DR

This paper develops interpolation problems for all $q$-Painlevé equations from $E_7^{(1)}$ to $(A_2+A_1)^{(1)}$, deriving their evolution equations, Lax pairs, and special solutions using the Padé interpolation method.

Contribution

It introduces a unified approach using Padé interpolation to derive key properties of all these $q$-Painlevé equations, including evolution equations and Lax pairs.

Findings

Derived evolution equations for all $q$-Painlevé types from $E_7^{(1)}$ to $(A_2+A_1)^{(1)}$

Obtained scalar Lax pairs for these equations

Constructed determinant formulae for special solutions

Abstract

We establish interpolation problems related to all the $q$-Painlev\'e equations of types from $E_7^{(1)}$ to $(A_2+A_1)^{(1)}$. By solving those problems, we can derive the evolution equations, the scalar Lax pairs and the determinant formulae of special solutions for the corresponding $q$-Painlev\'e equations.