Generalization of the Meijer G-function and the Fox–Wright function
In mathematics, the Fox H-functionH(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(((),()),(((-1,1/2)),()),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 was given by Ram Kishore Saxena.[3][4] A further generalization of this function, useful in physics and statistics, was provided by A.M. Mathai and Ram Kishore Saxena.[5][6]
References
^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.
^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.
Fox, Charles (1961), "The G and H functions as symmetrical Fourier kernels", Transactions of the American Mathematical Society, 98 (3): 395–429, doi:10.2307/1993339, ISSN 0002-9947, JSTOR 1993339, MR 0131578
Innayat-Hussain, AA (1987a), "New properties of hypergeometric series derivable from Feynman integrals. I: Transformation and reduction formulae", J. Phys. A: Math. Gen., 20 (13): 4109–4117, Bibcode:1987JPhA...20.4109I, doi:10.1088/0305-4470/20/13/019
Innayat-Hussain, AA (1987b), "New properties of hypergeometric series derivable from Feynman integrals. II: A generalization of the H-function", J. Phys. A: Math. Gen., 20 (13): 4119–4128, Bibcode:1987JPhA...20.4119I, 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. E. Horwood. ISBN 0-470-20010-3.