The Pad\'e interpolation method applied to additive difference Painlev\'e equations

TL;DR

This paper applies Padé interpolation on additive grids to derive equations, Lax pairs, and hypergeometric solutions for various additive difference Painlevé equations, advancing the analytical tools for these integrable systems.

Contribution

It introduces a novel application of Padé interpolation to additive difference Painlevé equations, providing explicit formulas and solutions.

Findings

Derived time evolution equations for $d$-Painlevé systems.

Obtained scalar Lax pairs of contiguous type.

Constructed determinant formulas for hypergeometric solutions.

Abstract

We study Pad\'e interpolation problems on an additive grid, related to additive difference ($d$-) Painlev\'e equations of type $E_7^{(1)}$, $E_6^{(1)}$, $D_4^{(1)}$ and $A_3^{(1)}$. By choosing suitable Pad\'e problems, we can derive time evolution equations, scalar Lax pairs of contiguous type and determinant formulae of special solutions given in terms of hypergeometric functions, for the corresponding $d$-Painlev\'e equations.