SIGMA 22 (2026), 040, 43 pages      arXiv:2503.11214      https://doi.org/10.3842/SIGMA.2026.040

Reformulation of $q$-Middle Convolution and Applications

Yumi Arai and Kouichi Takemura
Department of Mathematics, Ochanomizu University, 2-1-1 Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan

Received October 23, 2025, in final form April 11, 2026; Published online April 26, 2026

Abstract
We reformulate the $q$-convolution and the $q$-middle convolution introduced by Sakai and Yamaguchi, and we introduce $q$-analogues of the addition which is related to the gauge-transformation. A merit of the reformulation is the additivity on composition of two $q$-middle convolutions. We obtain sufficient conditions that the Jackson integrals associated with the $q$-convolution converge and satisfy the $q$-difference equation associated with the $q$-convolution. We present several third-order linear $q$-difference equations and solutions of them by using the $q$-middle convolution and the $q$-analogues of the addition.

Key words: hypergeometric function; $q$-hypergeometric equation; middle convolution; $q$-integral.

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