The Pad\'e interpolation method applied to $q$-Painlev\'e equations II (differential grid version)

TL;DR

This paper extends the application of Padé approximation methods to differential grids for deriving $q$-Painlevé equations of various types, providing new determinant formulas for special solutions and connecting to $q$-Appell Lauricella functions.

Contribution

It applies the Padé approximation method to differential grids for multiple $q$-Painlevé equations, deriving evolution equations, Lax pairs, and determinant formulas for solutions.

Findings

Derived $q$-Painlevé equations of types $E_6^{(1)}$, $D_5^{(1)}$, $A_4^{(1)}$, $(A_2+A_1)^{(1)}$ using Padé approximation.

Obtained determinant formulas for special solutions involving $q$-Appell Lauricella functions.

Extended the method to differential grids, complementing previous $q$-grid studies.

Abstract

Recently we studied Pad\'e interpolation problems of $q$-grid, related to $q$-Painlev\'e equations of type $E_7^{(1)}$, $E_6^{(1)}$, $D_5^{(1)}$, $A_4^{(1)}$ and $(A_2+A_1)^{(1)}$. By solving those problems, we could derive evolution equations, scalar Lax pairs and determinant formulae of special solutions for the corresponding $q$-Painlev\'e equations. It is natural that the $q$-Painlev\'e equations were derived by the interpolation method of $q$-grid, but it may be interesting in terms of differential grid that the Pad\'e interpolation method of differential grid (i.e. Pad\'e approximation method) has been applied to the $q$-Painlev\'e equation of type $D_5^{(1)}$ by Y. Ikawa. In this paper we continue the above study and apply the Pad\'e approximation method to the $q$-Painlev\'e equations of type $E_6^{(1)}$, $D_5^{(1)}$, $A_4^{(1)}$ and $(A_2+A_1)^{(1)}$. Moreover determinant formulae of the special solutions for $q$-Painlev\'e equation of type $E_6^{(1)}$ are given in terms of the terminating $q$-Appell Lauricella function.