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Two differential transforms involving the Gauss hypergeometric function in the kernels are considered. 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 of the first kind are proved. Special cases of products of cosine and sine functions are given. The results are established in terms of a generalized Lauricella function due to Srivastava and Daoust. Corresponding assertions for the Riemann–Liouville and the Erdélyi–Kober fractional integral transforms are presented. Statistical interpretations of fractional-order integrals and derivatives are also established.

Fractional differentiation of the product of Bessel functions of the first kind