Shoko Sasaki, Shun Takagi, Kouichi Takemura, “<nobr>$q$</nobr>-Middle Convolution and <nobr>$q$</nobr>-Painlevé Equation”, SIGMA, 2022, Volume 18,056, 21 pp.

This article is cited in 2 papers

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

Shoko Sasaki^a, Shun Takagi^a, 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

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.

Keywords: $q$-Painlevé equation, $q$-Heun equation, middle convolution, integral transformation.

MSC: 33E10, 34M55, 39A13

Received: January 31, 2022; in final form July 8, 2022; Published online July 20, 2022

Language: English

DOI: 10.3842/SIGMA.2022.056