Fractional di erentiation of the product of Bessel functions of the rst kind

Abstract

Two differential transforms involving the Gauss hypergeometric function in the kernels are consid-ered. They generalize the classical Riemann–Liouville and Erdélyi–Kober fractional differential operators.Formulas of compositions for such generalized fractional differentials with the product of Bessel functions ofthe rst kind are proved. Special cases of products of cosine and sine functions are given. The results are estab-lished in terms of a generalized Lauricella function due to Srivastava and Daoust. Corresponding assertionsfor the Riemann–Liouville and the Erdélyi–Kober fractional integral transforms are presented. Statistical in-terpretations of fractional-order integrals and derivatives are also established.