The Padé interpolation method applied to $q$-Painlevé equations II (differential grid version)

Abstract

Recently we studied Padé interpolation problems of $q$-grid, related to $q$-Painlevé 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 $q$-Painlevé equations. It is natural that the $q$-Painlevé equations were derived by the interpolation method of $q$-grid, but it may be interesting in terms of differential grid that the Padé interpolation method of differential grid (i.e. Padé approximation method) has been applied to the $q$-Painlevé equation of type $D_5^{(1)}$ by Y. Ikawa. In this paper we continue the above study and apply the Padé approximation method to the $q$-Painlevé 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 $q$-Painlevé equation of type $E_6^{(1)}$ are given in terms of the terminating $q$-Appell Lauricella function.

19 pages

References in corpus (4)

• Lax forms of the $q$-Painlevé equations
• Hypergeometric solutions to the q-Painlevé equation of type $A_4^{(1)}$
• A Lax Formalism for the Elliptic Difference Painlevé Equation
• The Padé interpolation method applied to $q$-Painlevé equations

Cited by in corpus (2)

• Variations of $q$-Garnier system
• A Variation of the $q$-Painlevé System with Affine Weyl Group Symmetry of Type $E_7^{(1)}$