Generalized fractional integration of Bessel function of the first kind

Two integral transforms involving the Gauss-hypergeometric function in the kernels are considered. They generalize the classical Riemann–Liouville and Erdélyi–Kober fractional integral operators. Formulas of compositions for such a generalized fractional integrals with Bessel function of the first kind are proved. Special cases of cosine and sine functions are given. The results are established in terms of generalized Wright function and generalized hypergeometric series. Corresponding assertions for Riemann–Liouville and Erdélyi–Kober fractional integral transforms are presented.