𝑞-Middle Convolution and 𝑞-Painlevé Equation

Abstract

A 𝑞-deformation of the middle convolution was introduced by Sakai and Yamaguchi. We apply it to a linear 𝑞-difference equation associated with the 𝑞-Painlevé VI equation. Then we obtain integral transformations. We investigate the 𝑞-middle convolution in terms of the affine Weyl group symmetry of the 𝑞-Painlevé VI equation. We deduce an integral transformation on the 𝑞-Heun equation.