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 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
− α ⋅ z
lim
α
α − β
α β
α + β
α β
α β
α − β
α − β
α β
lim
α − β
α β
β − α
α − β
α β
α β
α − β
α β
{\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}}}
is the complex conjugate of
[1] Compare to the Meijer G-function
2 π i
{\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 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.
