In mathematics, the Fox H-function H(x) is a generalization of the Meijer G-function and the Fox–Wright 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.

Plot of the Fox H function H((((a 1,α 1),...,(a n,α n)),((a n+1,α n+1),...,(a p,α p)),(((b 1,β 1),...,(b m,β m)),in ((b m+1,β m+1),...,(b q,β q))),z) with H(((),()),(((-1,½)),()),z)
Plot of the Fox H function H((((a 1,α 1),...,(a n,α n)),((a n+1,α n+1),...,(a p,α p)),(((b 1,β 1),...,(b m,β m)),in ((b m+1,β m+1),...,(b q,β q))),z) with H(((),()),(((-1,½)),()),z)

Relation to other Functions

Lambert W-function

A relation of the Fox H-Function to the -1 branch of the Lambert W-function is given by

where is the complex conjugate of .[1]

Meijer G-function

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 :[2]

A generalization of the Fox H-function is given by Ram Kishore Saxena.[3][4] For a further generalization of this function, useful in physics and statistics was given by A.M.Mathai and Ram Kishore Saxena,[5][6]

References

  1. Rathie and Ozelim, Pushpa Narayan and Luan Carlos de Sena Monteiro. "On the Relation between Lambert W-Function and Generalized Hypergeometric Functions". Researchgate. Retrieved 1 March 2023.
  2. (Srivastava & Manocha 1984, p. 50)
  3. Mathai, A. M.; Saxena, R. K.; Saxena, Ram Kishore (1973). Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences. Springer. ISBN 978-0-387-06482-6.
  4. Innayat-Hussain (1987a)
  5. Mathai, A. M.; Saxena, Rajendra Kumar (1978). The H-function with Applications in Statistics and Other Disciplines. Wiley. ISBN 978-0-470-26380-8.
  6. Rathie (1997)
  • 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. E. Horwood. ISBN 0-470-20010-3.


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