Fox H-function

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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. Another generalization of 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).

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


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  • 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 
  • 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 
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  • 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 691138 
  • Srivastava, H. M.; Manocha, H. L. (1984). A treatise on generating functions. ISBN 0-470-20010-3.