On generalized composite fractional q-derivative

Abstract

In the present paper, we define a generalized composite fractional q-derivative D^{\alpha,\beta,\nu}_q and obtain some results for it. These results are image of power function under D^{\alpha,\beta,\nu}_q,  composition of Riemann-Liouville type fractional q-integral I^\alpha_q with D^{\alpha,\beta,\nu}_q  and q-Laplace transform of D^{\alpha,\beta,\nu}_q. We also obtain solution of a q-difference equation with derivative as D^{\alpha,\beta,\nu}_q  and discuss some special cases. A q-difference equation of D^{\alpha,\beta,\nu}_q  is solved using q-Laplace transform and its inverse.