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On the Mellin transform of a product of hypergeometric functions. (English) Zbl 0915.33006

We obtain representations for the Mellin transform of the product of generalized hypergeometric functions \({}_0F_1 [-a^2x^2] {}_1F_2[-b^2x^2]\) 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 (e.g., integrals involving products of powers, Bessel and Lommel functions), it contains numerous integrals on 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 \({}_3F_2[1]\). Furthermore, we give the sine and cosine transforms of \({}_1F_2 [-b^2x^2]\).

MSC:

33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
33C20 Generalized hypergeometric series, \({}_pF_q\)
44A20 Integral transforms of special functions
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