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On the H-Function With Applications

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The H-Function

Abstract

Mellin–Barnes integrals are discovered by Salvatore Pincherle, an Italian mathematician in the year 1888. These integrals are based on the duality principle between linear differential equations and linear difference equations with rational coefficients. The theory of these integrals has been developed by Mellin (1910) and has been used in the development of the theory of hypergeometric functions by Barnes (1908). Important contributions of Salvatore Pincherle are recently given in a paper by Mainardi and Pagnini (2003). In the year 1946, these integrals were used by Meijer to introduce the G-function into mathematical analysis. From 1956 to 1970 lot of work has been done on this function, which can be seen from the bibliography of the book by Mathai and Saxena (1973a).

In the year 1961, in an attempt to discover a most generalized symmetrical Fourier kernel, Charles Fox (1961) defined a new function involving Mellin–Barnes integrals, which is a generalization of the G-function of Meijer. This function is called Fox’s H-function or the H-function. The importance of this function is realized by the scientists, engineers and statisticians due to its vast potential of its applications in diversified fields of science and engineering. This function includes, among others, the functions considered by Boersma (1962), Mittag-Leffler (1903), generalized Bessel function due to Wright (1934), the generalization of the hypergeometric functions studied by Fox (1928), and Wright (1935, 1940), Krätzel function (Krätzel 1979), generalized Mittag-Leffler function due to Dzherbashyan (1960), generalized Mittag-Leffler function due to Prabhakar (1971) and multi-index Mittag-Leffler function due to Kiryakova (2000), etc. Except the functions of Boersma (1962), the aforesaid functions cannot be obtained as special cases of the G-function of Meijer (1946), hence a study of the H-function will cover wider range than the G-function and gives general, deeper, and useful results directly applicable in various problems of physical, biological, engineering and earth sciences, such as fluid flow, rheology, diffusion in porous media, kinematics in viscoelastic media, relaxation and diffusion processes in complex systems, propagation of seismic waves, anomalous diffusion and turbulence, etc. see, Caputo (1969), Glöckle and Nonnenmacher (1993), Mainardi et al. (2001), Saichev and Zaslavsky (1997), Hilfer (2000), Metzler and Klafter (2000), Podlubny (1999), Schneider (1986) and Schneider and Wyss (1989) and others.

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Authors and Affiliations

  1. Centre for Mathematical Sciences (CMS), Arunapuram P.O., Pala, 686574, Pala Campus, India

    Prof. Dr. A. M. Mathai

  2. 34 Panchi Batti Chauraha, Jodhpur, 342 011, Ratananda, India

    Prof. Dr. Ram Kishore Saxena

  3. United Nations Vienna International Centre Space Application Programme, 1400, Wien, Austria

    Prof. Dr. Hans J. Haubold

Authors
  1. Prof. Dr. A. M. Mathai
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  2. Prof. Dr. Ram Kishore Saxena
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  3. Prof. Dr. Hans J. Haubold
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Correspondence to Hans J. Haubold .

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Mathai, A.M., Saxena, R.K., Haubold, H.J. (2010). On the H-Function With Applications. In: The H-Function. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-0916-9_1

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