Spectral Types of Linear $q$-Difference Equations and $q$-Analog of Middle Convolution

We present a $q$-analog of the middle convolution for linear $q$-difference equations with rational coefficients. In the differential case, the middle convolution was defined by Katz, who examined its properties in detail. In this paper, we define a $q$-analog of the middle convolution. Moreover, we show that it can also be expressed as a $q$-analog of the Euler transformation. The $q$-middle convolution transforms a Fuchsian equation to a Fuchsian equation and preserves the rigidity index of $q$-difference equations.