Fractional q-symmetric calculus on a time scale

Abstract In this paper, the definitions of q-symmetric exponential function and q-symmetric gamma function are presented. By a q-symmetric exponential function, we shall illustrate the Laplace transform method and define and solve several families of linear fractional q-symmetric difference equations with constant coefficients. We also introduce a q-symmetric analogue Mittag-Leffler function and study q-symmetric Caputo fractional initial value problems. It is hoped that our work will provide foundation and motivation for further studying of fractional q-symmetric difference systems