Fractional q-leibniz rule and applications

Abstract

This chapter includes analytic investigations on q-type Leibniz rules of q-Riemann-Liouville fractional operator introduced by Al-Salam and Verma in (Pac. J. Math. 60(2), 1-9, 1975). In this chapter, we provide a generalization of the Riemann-Liouville fractional q-Leibniz formula introduced by Agarwal in (Ganita 27(1-2), 25-32, 1976). Purohit (Kyungpook Math. J. 50(4), 473-482, 2010) introduced a Leibniz formula for Weyl q-fractional operator only when α is an integer. In this respect, We extend Purohit's result for any. We end the chapter with deriving some q-series and formulae by applying the fractional Leibniz formula mentioned and derived earlier in the chapter. © 2012 Springer-Verlag Berlin Heidelberg.