Fox H-function

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

H_{{p,q}}^{{\,m,n}}\!\left[z\left|{\begin{matrix}(a_{1},A_{1})&(a_{2},A_{2})&\ldots &(a_{p},A_{p})\\(b_{1},B_{1})&(b_{2},B_{2})&\ldots &(b_{q},B_{q})\end{matrix}}\right.\right]={\frac {1}{2\pi i}}\int _{L}{\frac {(\prod _{{j=1}}^{m}\Gamma (b_{j}+B_{j}s))(\prod _{{j=1}}^{n}\Gamma (1-a_{j}-A_{j}s))}{(\prod _{{j=m+1}}^{q}\Gamma (1-b_{j}-B_{j}s))(\prod _{{j=n+1}}^{p}\Gamma (a_{j}+A_{j}s))}}z^{{-s}}\,ds

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 A_j = B_k = C, C > 0 for j = 1...p and k = 1...q (Srivastava 1984, p. 50):

H_{{p,q}}^{{\,m,n}}\!\left[z\left|{\begin{matrix}(a_{1},C)&(a_{2},C)&\ldots &(a_{p},C)\\(b_{1},C)&(b_{2},C)&\ldots &(b_{q},C)\end{matrix}}\right.\right]={\frac {1}{C}}G_{{p,q}}^{{\,m,n}}\!\left(\left.{\begin{matrix}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{matrix}}\;\right|\,z^{{1/C}}\right).

References

Fox, Charles (1961), "The G and H functions as symmetrical Fourier kernels", Transactions of the American Mathematical Society, 98: 395–429, ISSN 0002-9947, JSTOR 1993339, MR 0131578, doi:10.2307/1993339
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
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 513025
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.

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