Fox H-function

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

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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