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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.MSC: 92B20; 68T05; 39A11; 34K13

The q-symmetric analogs of Cauchy's formulas for multiple integrals are obtained. We introduce the concepts of the fractional q-symmetric integrals and fractional q-symmetric derivatives and discuss some of their properties. By using some properties of q-symmetric fractional integrals and fractional difference operators, we study a boundary value problem with nonlocal boundary conditions. MSC: 26A33; 34B15

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