Fox H-function

From Wikipedia, the free encyclopedia
  (Redirected from H-Function)
Jump to navigation Jump to search

In mathematics, the Fox H-function H(x) is a generalization of the Meijer G-function introduced by Charles Fox (1961). It is defined by a Mellin–Barnes integral

where L is a certain contour separating the poles of the two factors in the numerator. Compare to the Meijer G-function,

The special case for which the Fox H reduces to the Meijer G is Aj = Bk = C, C > 0 for j = 1...p and k = 1...q (Srivastava 1984, p. 50):

A generalization of the Fox H-function is given by Innayat Hussain AA (1987). For a further generalization of this function, useful in physics and statistics, see Rathie (1997).


  • Fox, Charles (1961), "The G and H functions as symmetrical Fourier kernels", Transactions of the American Mathematical Society, 98: 395–429, doi:10.2307/1993339, ISSN 0002-9947, JSTOR 1993339, MR 0131578
  • Innayat-Hussain, AA (1987), "New properties of hypergeometric series derivable from Feynman integrals. I: Transformation and reduction formulae", J. Phys. A: Math. Gen., 20: 4109–4117, doi:10.1088/0305-4470/20/13/019
  • Innayat-Hussain, AA (1987), "New properties of hypergeometric series derivable from Feynman integrals. II: A generalization of the H-function", J. Phys. A: Math. Gen., 20: 4119–4128, doi:10.1088/0305-4470/20/13/020
  • Kilbas, Anatoly A. (2004), H-Transforms: Theory and Applications, CRC Press, ISBN 978-0415299169
  • Mathai, A. M.; Saxena, Ram Kishore (1978), The H-function with applications in statistics and other disciplines, Halsted Press [John Wiley & Sons], New York-London-Sidney, ISBN 978-0-470-26380-8, MR 0513025
  • Mathai, A. M.; Saxena, Ram Kishore; Haubold, Hans J. (2010), The H-function, Berlin, New York: Springer-Verlag, ISBN 978-1-4419-0915-2, MR 2562766
  • Rathie, Arjun K. (1997), "A new generalization of generalized hypergeometric function", Le Matematiche, LII: 297–310.
  • Srivastava, H. M.; Gupta, K. C.; Goyal, S. P. (1982), The H-functions of one and two variables, New Delhi: South Asian Publishers Pvt. Ltd., MR 0691138
  • Srivastava, H. M.; Manocha, H. L. (1984). A treatise on generating functions. ISBN 0-470-20010-3.