On the Mellin transform of a product of hypergeometric functions

Allen R. Miller
1616 Eighteenth Street NW
Washington, D. C. 20009-2530
USA
and
H. M. Srivastava
Department of Mathematics and Statistics
University of Victoria
Victoria, B. C. V8W 3P4
Canada.

Abstract:

We obtain representations for the Mellin transform of the product of generalized hypergeometric functions ₀F₁[-a²x²]₁F₂[-b²x²] for a,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 for ₃F₂[1]. Furthermore, we give the sine and cosine transforms of ₁F₂[-b²x²].