Inverse operators, \(q\)-fractional integrals, and \(q\)-Bernoulli polynomials
Let \(I_{\alpha}f\) be the fractional integral of order \(\alpha\) of a function \(f\) which is integrable and of period \(2\pi\). The definition and properties of the operator \(I_{\alpha}\) can be found in \textit{A. Zygmund}'s book [Trigonometric series. 2nd ed., Vol. 2, Cambridge Univ. Press, Cambridge (1959; Zbl 0085.05601)]. In this paper the authors define a \(q\)-analogue of the fractional integral operators \(I_{\alpha}\), so that for \(\alpha=1\) it becomes a right inverse of the Askey-Wilson operator \({\mathcal D}_{q}\). The operator \({\mathcal D}_{q}\) was introduced in [\textit{R. Askey} and \textit{J. Wilson}, Mem. Am. Math. Soc. 319, 55 p. (1985; Zbl 0572.33012)] and is a \(q\)-analogue of the differentiation operator. Then the authors establish the semigroup property for \(q\)-fractional integrals and derivatives. They study properties of the kernel of \(q\)-fractional integrals and show how they give rise to a \(q\)-analogue of Bernoulli polynomials, which are now polynomials of two variables \(x\) and \(y\). As \(q \rightarrow 1\) the polynomials become polynomials in \(x-y\), a convolution kernel in one variable. They 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.
scientific article; zbMATH DE number 1763672
scientific article