Fox H-function

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title: "Fox H-function" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["hypergeometric-functions", "special-functions"] description: "Generalization of the Meijer G-function and the Fox–Wright function" topic_path: "general/hypergeometric-functions" source: "https://en.wikipedia.org/wiki/Fox_H-function" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Generalization of the Meijer G-function and the Fox–Wright 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. ::figure[src="https://upload.wikimedia.org/wikipedia/commons/6/62/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" caption="2}})),()),z)"] ::

Relation to other functions

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| \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 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).

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.

References

• {{Citation | last1= Kilbas | first1= Anatoly A. | title=H-Transforms: Theory and Applications | publisher= CRC Press | isbn= 978-0415299169 | year= 2004 }}

• .

References

1. Rathie and Ozelim, Pushpa Narayan and Luan Carlos de Sena Monteiro. "On the Relation between Lambert W-Function and Generalized Hypergeometric Functions".
2. {{harv. Srivastava. Manocha. 1984
3. (1973). "Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences". Springer.
4. {{harvtxt. Innayat-Hussain. 1987a
5. (1978). "The H-function with Applications in Statistics and Other Disciplines". Wiley.
6. {{harvtxt. Rathie. 1997

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