In mathematics, the Fox H-function H(x) is a generalization of the Meijer G-function and the Fox–Wright function introduced by txt. 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.
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| < \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.[1]
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 > 0 for j = 1...p and k = 1...q :[2]
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.[3][4] A further generalization of this function, useful in physics and statistics, was provided by A.M. Mathai and Ram Kishore Saxena.[5][6]
References
- ^ Rathie and Ozelim, Pushpa Narayan and Luan Carlos de Sena Monteiro. "On the Relation between Lambert W-Function and Generalized Hypergeometric Functions". Researchgate. Retrieved 1 March 2023.
- ^ (Srivastava & Manocha 1984, p. 50)
- ^ Mathai, A. M.; Saxena, R. K.; Saxena, Ram Kishore (1973). Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences. Springer. ISBN 978-0-387-06482-6.
- ^ Innayat-Hussain (1987a)
- ^ Mathai, A. M. & Saxena, Rajendra Kumar (1978). The H-function with Applications in Statistics and Other Disciplines. Wiley. ISBN 978-0-470-26380-8.
- ^ Rathie (1997)
- Fox, Charles (1961), "The G and H functions as symmetrical Fourier kernels", Transactions of the American Mathematical Society. 98 (3): 395–429, doi:10.2307/1993339. ISSN 0002-9947. JSTOR 1993339. MR 0131578
- Innayat-Hussain, AA (1987a), "New properties of hypergeometric series derivable from Feynman integrals. I: Transformation and reduction formulae", J. Phys. A: Math. Gen.. 20 (13): 4109–4117, Bibcode:1987JPhA...20.4109I. doi:10.1088/0305-4470/20/13/019
- Innayat-Hussain, AA (1987b), "New properties of hypergeometric series derivable from Feynman integrals. II: A generalization of the H-function", J. Phys. A: Math. Gen.. 20 (13): 4119–4128, Bibcode:1987JPhA...20.4119I. doi:10.1088/0305-4470/20/13/020
- Kilbas, Anatoly A. (2004), H-Transforms: Theory and Applications, CRC Press, ISBN 978-0415299169
- Mathai, A. M. & Saxena, Ram Kishore (1978), The H-function with applications in statistics and other disciplines, Halsted Press [John Wiley & Sons], New York-London-Sidney, ISBN 978-0-470-26380-8. MR 513025
- Mathai, A. M.; Saxena, Ram Kishore; Haubold, Hans J. (2010), The H-function, Berlin, New York: Springer-Verlag, ISBN 978-1-4419-0915-2. MR 2562766
- Rathie, Arjun K. (1997), "A new generalization of generalized hypergeometric function", Le Matematiche. LII: 297–310.
- Srivastava, H. M.; Gupta, K. C.; Goyal, S. P. (1982), "The H-functions of one and two variables", New Delhi: South Asian Publishers Pvt. Ltd., MR 691138
- Srivastava, H. M. & Manocha, H. L. (1984). A treatise on generating functions. E. Horwood. ISBN 0-470-20010-3.