Fox H-function

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} [z | \begin{matrix} (a_{1}, A_{1}) & (a_{2}, A_{2}) & \dots & (a_{p}, A_{p}) \\ (b_{1}, B_{1}) & (b_{2}, B_{2}) & \dots & (b_{q}, B_{q}) \end{matrix}] = \frac{1}{2 π i} \int_{L} \frac{\prod_{j = 1}^{m} Γ (b_{j} + B_{j} s) \prod_{j = 1}^{n} Γ (1 - a_{j} - A_{j} s)}{\prod_{j = m + 1}^{q} Γ (1 - b_{j} - B_{j} s) \prod_{j = n + 1}^{p} Γ (a_{j} + A_{j} s)} z^{- s} d s,

where L is a certain contour separating the poles of the two factors in the numerator.

Relation to other functions

Meijer G-function

Compare to the Meijer G-function

$G_{p, q}^{m, n} (\begin{matrix} a_{1}, \dots, a_{p} \\ b_{1}, \dots, b_{q} \end{matrix} | z) = \frac{1}{2 π i} \int_{L} \frac{\prod_{j = 1}^{m} Γ (b_{j} - s) \prod_{j = 1}^{n} Γ (1 - a_{j} + s)}{\prod_{j = m + 1}^{q} Γ (1 - b_{j} + s) \prod_{j = n + 1}^{p} Γ (a_{j} - s)} z^{s} d s .$

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

Lambert W-function

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

$\overline{W_{- 1} (- α z)} = lim_{β \to α^{-}} {\begin{matrix} \frac{α^{2}}{β} {((α - β) z)}^{\frac{α}{β}} H_{1, 2}^{1, 1} (- {((α - β) z)}^{\frac{α}{β} - 1} | \begin{matrix} (\frac{α + β}{β}, \frac{α}{β}) \\ (0, 1), (- \frac{α}{β}, \frac{α - β}{β}) \end{matrix}), if | z | < \frac{1}{e | α |} \\ \frac{α^{2}}{β} {((α - β) z)}^{- \frac{α}{β}} H_{2, 1}^{1, 1} (- {((α - β) z)}^{1 - \frac{α}{β}} | \begin{matrix} (1, 1), (\frac{β - α}{β}, \frac{α - β}{β}) \\ (- \frac{α}{β}, \frac{α}{β}) \end{matrix}), else \end{matrix}$

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.

References

↑ (Srivastava & Manocha 1984, p. 50)
↑ 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.
↑ 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 – via Internet Archive.
↑ 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 0513025
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 0691138
Srivastava, H. M.; Manocha, H. L. (1984). A treatise on generating functions. E. Horwood. ISBN 0-470-20010-3.

External links

[1] (Srivastava & Manocha 1984, p. 50)

[2] 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.

[3] 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.

[4] Innayat-Hussain (1987a)

[5] Mathai, A. M.; Saxena, Rajendra Kumar (1978). The H-function with Applications in Statistics and Other Disciplines. Wiley. ISBN 978-0-470-26380-8 – via Internet Archive.

[6] Rathie (1997)