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Some New Facts Concerning the Delta Neutral Case of Fox’s H Function

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Abstract

In this paper, we find several new properties of a class of Fox’s H functions which we call delta neutral. In particular, we find an expansion in the neighborhood of the finite non-zero singularity and give new Mellin transform formulas under a special restriction on parameters. The last result is applied to prove a conjecture regarding the representing measure for gamma ratio in Bernstein’s theorem. Furthermore, we find the weak limit of measures expressed in terms of the H function which furnishes a regularization method for integrals containing the delta neutral and zero-balanced cases of Fox’s H function. We apply this result to extend a recently discovered integral equation to the zero-balanced case. In the last section of the paper, we consider a reduced form of this integral equation for Meijer’s G function. This leads to certain expansions believed to be new even in the case of the Gauss hypergeometric function.

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Acknowledgments

This work was supported by the Russian Science Foundation under Project 14-11-00022.

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Authors and Affiliations

  1. Far Eastern Federal University, 8 Sukhanova Street, Vladivostok, 690950, Russia

    D. B. Karp & E. G. Prilepkina

  2. Institute of Applied Mathematics, FEBRAS, 7 Radio Street, Vladivostok, 690041, Russia

    D. B. Karp & E. G. Prilepkina

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  1. D. B. Karp
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  2. E. G. Prilepkina
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Correspondence to D. B. Karp.

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Communicated by Henrik Laurberg Pedersen.

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Karp, D.B., Prilepkina, E.G. Some New Facts Concerning the Delta Neutral Case of Fox’s H Function. Comput. Methods Funct. Theory 17, 343–367 (2017). https://doi.org/10.1007/s40315-016-0183-x

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