Study of $q$-Garnier system by Pad\'e method

Abstract

We give a simple form of the evolution equation and a scalar Lax pair for the $q$-Garnier system. Some reductions to the $q$-Painlev\'e equations and the autonomous case as a generalized QRT system are discussed. Using two kinds of Pad\'e problems on differential grid and $q$-grid, we derive some special solutions of the $q$-Garnier system in terms of the $q$-Appell Lauricella function and the generalized $q$-hypergeometric function.