$q$-Middle Convolution and $q$-Painlev\'e Equation
Shoko Sasaki, Shun Takagi, Kouichi Takemura
TL;DR
This paper explores a $q$-deformation of middle convolution and its application to the $q$-Painlevé VI equation, revealing new integral transformations and symmetries related to affine Weyl groups.
Contribution
It introduces a novel application of $q$-middle convolution to $q$-difference equations, connecting it with affine Weyl group symmetries and deriving new integral transformations.
Findings
Derived integral transformations for $q$-Painlevé VI.
Connected $q$-middle convolution with affine Weyl group symmetry.
Obtained an integral transformation on the $q$-Heun equation.
Abstract
A -deformation of the middle convolution was introduced by Sakai and Yamaguchi. We apply it to a linear -difference equation associated with the -Painlev\'e VI equation. Then we obtain integral transformations. We investigate the -middle convolution in terms of the affine Weyl group symmetry of the -Painlev\'e VI equation. We deduce an integral transformation on the -Heun equation.
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