Reformulation of $q$-middle convolution and applications
Yumi Arai, Kouichi Takemura
TL;DR
The paper presents a streamlined reformulation of Sakai–Yamaguchi's q–convolution and q–middle convolution, introducing a q–deformation of addition and establishing convergence criteria for Jackson-integral representations. A central advance is the additivity of composition: mc^q_lambda ∘ mc^q_mu = mc^q_{lambda+mu} under suitable conditions, enabling systematic generation of higher-order q–difference equations from simpler seeds. The authors construct and analyze several third-order q–difference equations, including generalized q–hypergeometric and q–Jordan–Pochhammer variants, and provide formal q–integral representations that, under convergence hypotheses, yield actual solutions. They also develop a framework for composing q–middle convolutions and relate these constructions to the spectral-type/rigidity viewpoint, with q→1 limits connecting to classical differential equations. Overall, the work supplies new tools for generating and solving q–difference equations via q-convolution and adds depth to the intersection of q-analogues and rigid local systems.
Abstract
We reformulate the $q$-convolution and the $q$-middle convolution introduced by Sakai and Yamaguchi, and we introduce $q$-deformations 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 $q$-deformations of the addition.