Inverse Operators, q-Fractional Integrals, and q-Bernoulli Polynomials

Abstract

We introduce operators of q-fractional integration through inverses of the Askey-Wilson operator and use them to introduce a q-fractional calculus. We establish the semigroup property for fractional integrals and fractional derivatives. We study properties of the kernel of q-fractional integral and show how they give rise to a q-analogue of Bernoulli polynomials, which are now polynomials of two variables, x and y. As q->1 the polynomials become polynomials in x-y, a convolution kernel in one variable. We also evaluate explicitly a related kernel of a right inverse of the Askey-Wilson operator on an L^2 space weighted by the weight function of the Askey-Wilson polynomials.