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.
Highlights
The q-calculus is not of recent appearance
It was initiated in the twenties of the last century. It has gained considerable popularity and importance during the last three decades or so. This is due to its distinguished applications in numerous diverse fields of physics such as cosmic strings and black holes [ ], conformal quantum mechanics [ ], nuclear and high energy physics [ ], just to name a few
A qLaplace transform method has been developed by Abdi [ ] and applied to q-difference equations [, ]
Summary
The q-calculus is not of recent appearance. It was initiated in the twenties of the last century. We introduced basic concepts of fractional q-symmetric integral and derivative operators in [ ]. For a function f : T −→ R, the q-symmetric derivative is defined by f (qx) – f (q– x). (Dqf )( ) := f ( ), and the q-symmetric derivatives of higher order are defined by. For operators defined in this manner, the following is valid:. The q-symmetric factorial is defined in the following way. Define the q-symmetric gamma function by q– t α–. The fractional q-symmetric integration has the following semigroup property: Iqα, Iqβ, f (x) = Iqα,+ β f (x). ([ ]) For α ∈ R+, the following identity is valid: Iqα, f (x) = Iqα,+ Dqf (x) +. We define the fractional q-symmetric derivative of Riemann-Liouville type of a function f (x) by.
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