Fox H-function

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

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

where L is a certain contour separating the poles of the two factors in the numerator.

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

${\displaystyle {\overline {\operatorname {W} _{-1}\left(-\alpha \cdot z\right)}}={\begin{cases}\lim _{\beta \to \alpha ^{-}}\left[{\frac {\alpha ^{2}\cdot \left(\left(\alpha -\beta \right)\cdot z\right)^{\frac {\alpha }{\beta }}}{\beta }}\cdot \operatorname {H} _{1,\,2}^{1,\,1}\left({\begin{matrix}\left({\frac {\alpha +\beta }{\beta }},\,{\frac {\alpha }{\beta }}\right)\\\left(0,\,1\right),\,\left(-{\frac {\alpha }{\beta }},\,{\frac {\alpha -\beta }{\beta }}\right)\\\end{matrix}}\mid -\left(\left(\alpha -\beta \right)\cdot z\right)^{{\frac {\alpha }{\beta }}-1}\right)\right],\,{\text{for}}\left|z\right|<{\frac {1}{e\left|\alpha \right|}}\\\lim _{\beta \to \alpha ^{-}}\left[{\frac {\alpha ^{2}\cdot \left(\left(\alpha -\beta \right)\cdot z\right)^{-{\frac {\alpha }{\beta }}}}{\beta }}\cdot \operatorname {H} _{2,\,1}^{1,\,1}\left({\begin{matrix}\left(1,\,1\right),\,\left({\frac {\beta -\alpha }{\beta }},\,{\frac {\alpha -\beta }{\beta }}\right)\\\left(-{\frac {\alpha }{\beta }},\,{\frac {\alpha }{\beta }}\right)\\\end{matrix}}\mid -\left(\left(\alpha -\beta \right)\cdot z\right)^{1-{\frac {\alpha }{\beta }}}\right)\right],\,{\text{otherwise}}\\\end{cases}}}$where ${\displaystyle {\overline {z}}}$ is the complex conjugate of ${\displaystyle z}$.^[1]

Compare to the Meijer G-function

${\displaystyle G_{p,q}^{\,m,n}\!\left(\left.{\begin{matrix}a_{1},\dots ,a_{p}\\b_{1},\dots ,b_{q}\end{matrix}}\;\right|\,z\right)={\frac {1}{2\pi i}}\int _{L}{\frac {\prod _{j=1}^{m}\Gamma (b_{j}-s)\,\prod _{j=1}^{n}\Gamma (1-a_{j}+s)}{\prod _{j=m+1}^{q}\Gamma (1-b_{j}+s)\,\prod _{j=n+1}^{p}\Gamma (a_{j}-s)}}\,z^{s}\,ds.}$

The special case for which the Fox H reduces to the Meijer G is A_j = B_k = C, C > 0 for j = 1...p and k = 1...q :^[2]

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

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]

• 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.

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• Use in solving ${\displaystyle x+x^{a}=y}$ on MathOverflow