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Fox H-function

Generalization of the Meijer G-function and the Fox-Wright function

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Source updated September 16, 2026

In mathematics, the Fox H-function $H(z)$ is a generalization of the Meijer G-function and the Fox-Wright function introduced by Charles Fox (1961). It is defined by the 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.

01Relation to other functions

Meijer G-function

Compare to the Meijer G-function

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

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 :

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

Lambert W-function

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

${\overline {W_{-1}\left(-\alpha z\right)}}=\lim _{\beta \to \alpha ^{-}}{\begin{cases}{\frac {\alpha ^{2}}{\beta }}\left(\left(\alpha -\beta \right)z\right)^{{\frac {\alpha }{\beta }}{\hphantom {-}}}H_{1,2}^{1,1}{\Bigg (}-\left(\left(\alpha -\beta \right)z\right)^{{\frac {\alpha }{\beta }}-1}{\Bigg |}{\begin{matrix}{\big (}{\frac {\alpha +\beta }{\beta }},\,{\frac {\alpha }{\beta }}{\big )}\\{\big (}0,\,1{\big )},\,{\big (}-{\frac {\alpha }{\beta }},\,{\frac {\alpha -\beta }{\beta }}{\big )}\end{matrix}}{\Bigg )},\,{\text{if}}\left|z\right|<{\frac {1}{e\left|\alpha \right|}}\\{\frac {\alpha ^{2}}{\beta }}\left(\left(\alpha -\beta \right)z\right)^{-{\frac {\alpha }{\beta }}}H_{2,1}^{1,1}{\Bigg (}-\left(\left(\alpha -\beta \right)z\right)^{1-{\frac {\alpha }{\beta }}}{\Bigg |}{\begin{matrix}{\big (}1,\,1{\big )},\,{\big (}{\frac {\beta -\alpha }{\beta }},\,{\frac {\alpha -\beta }{\beta }}{\big )}\\{\big (}-{\frac {\alpha }{\beta }},\,{\frac {\alpha }{\beta }}{\big )}\end{matrix}}{\Bigg )},\,{\text{else}}\\\end{cases}}$

where ${\overline {z}}$ is the complex conjugate of $z$ .

Generalizations

A generalization of the Fox H-function was given by Ram Kishore Saxena. A further generalization of this function, useful in physics and statistics, was provided by A.M. Mathai and Ram Kishore Saxena.

Watch videos about Fox H-functionExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Fox H-function”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

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).svg by WalkingRadiance, CC BY-SA 4.0

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