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

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.