Reformulation of $q$-Middle Convolution and Applications

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.