### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 056, 21 pages      arXiv:2201.03960      https://doi.org/10.3842/SIGMA.2022.056

### $q$-Middle Convolution and $q$-Painlevé Equation

Shoko Sasaki a, Shun Takagi a and Kouichi Takemura b
a) Department of Mathematics, Faculty of Science and Engineering, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan
b) Department of Mathematics, Ochanomizu University, 2-1-1 Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan

Received January 31, 2022, in final form July 08, 2022; Published online July 20, 2022

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.

Key words: $q$-Painlevé equation; $q$-Heun equation; middle convolution; integral transformation.

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