On generalized composite fractional q-derivative

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