On the Mellin transform of a product of hypergeometric functions

Abstract

AbstractWe obtain representations for the Mellin transform of the product of generalized hypergeometric functions0F1[−a2x2]1F2[−b2x2]fora, b > 0. The later transform is a generalization of the discontinuous integral of Weber and Schafheitlin; in addition to reducing to other known integrals (for example, integrals involving products of powers, Bessel and Lommel functions), it contains numerous integrals of interest that are not readily available in the mathematical literature. As a by-product of the present investigation, we deduce the second fundamental relation for3F2[1]. Furthermore, we give the sine and cosine transforms of1F2[−b2x2].