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. 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_js) \, \prod_{j=1}^n\Gamma(1-a_j-A_js)} {\prod_{j=m+1}^q\Gamma(1-b_j-B_js) \, \prod_{j=n+1}^p\Gamma(a_j+A_js)} z^{-s} \, ds, $$ where L is a certain contour separating the poles of the two factors in the numerator.

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

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

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 Aj = Bk = C, C &gt; 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). $$

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.