Reformulation of -Middle Convolution and Applications
arXiv:2503.11214 · doi:10.3842/SIGMA.2026.040
Abstract
We reformulate the -convolution and the -middle convolution introduced by Sakai and Yamaguchi, and we introduce -analogues of the addition which is related to the gauge-transformation. A merit of the reformulation is the additivity on composition of two -middle convolutions. We obtain sufficient conditions that the Jackson integrals associated with the -convolution converge and satisfy the -difference equation associated with the -convolution. We present several third-order linear -difference equations and solutions of them by using the -middle convolution and the -analogues of the addition.