Fox H-function

Short description: Generalization of the Meijer G-function and the Fox–Wright function

"H function" redirects here. It is not to be confused with Harmonic number.

In mathematics, the Fox H-function H(x) is a generalization of the Meijer G-function and the Fox–Wright function introduced by Charles Fox (1961). It is defined by a Mellin–Barnes integral

$H_{p,q}^{m,n}\left[z|\begin{array}{cccc}(a_1,A_1)& (a_2,A_2)& \dots & (a_p,A_p)\\ (b_1,B_1)& (b_2,B_2)& \dots & (b_q,B_q)\end{array}\right]=\frac{1}{2\pi i}\underset{L}{\int }\frac{\prod _{j=1}^m\mathrm{\Gamma }(b_j+B_{j}s)\prod _{j=1}^n\mathrm{\Gamma }(1-a_j-A_{j}s)}{\prod _{j=m+1}^q\mathrm{\Gamma }(1-b_j-B_{j}s)\prod _{j=n+1}^p\mathrm{\Gamma }(a_j+A_{j}s)}{z}^{-s}ds,$

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

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(((),()),(((-1,⁠1/2⁠)),()),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

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

Meijer G-function

Compare to the Meijer G-function

$G_{p,q}^{m,n}\left(\begin{array}{c}{a}_1,\dots ,a_p\\ b_1,\dots ,b_q\end{array}|z\right)=\frac{1}{2\pi i}\underset{L}{\int }\frac{\prod _{j=1}^m\mathrm{\Gamma }(b_j-s)\prod _{j=1}^n\mathrm{\Gamma }(1-a_j+s)}{\prod _{j=m+1}^q\mathrm{\Gamma }(1-b_j+s)\prod _{j=n+1}^p\mathrm{\Gamma }(a_j-s)}{z}^{s}ds.$

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 :

$H_{p,q}^{m,n}\left[z|\begin{array}{cccc}(a_1,C)& (a_2,C)& \dots & (a_p,C)\\ (b_1,C)& (b_2,C)& \dots & (b_q,C)\end{array}\right]=\frac{1}{C}{G}_{p,q}^{m,n}\left(\begin{array}{c}{a}_1,\dots ,a_p\\ b_1,\dots ,b_q\end{array}|z^{1/C}\right).$

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

• 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 
• 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, doi:10.1088/0305-4470/20/13/019, Bibcode: 1987JPhA...20.4109I 
• 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, doi:10.1088/0305-4470/20/13/020, Bibcode: 1987JPhA...20.4119I 
• 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 
• Mathai, A. M.; Saxena, Ram Kishore; Haubold, Hans J. (2010), The H-function, Berlin, New York: Springer-Verlag, ISBN 978-1-4419-0915-2 
• 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. 
• Srivastava, H. M.; Manocha, H. L. (1984). A treatise on generating functions. E. Horwood. ISBN 0-470-20010-3. 

External links

• hypergeom on GitLab
• Use in solving $x+x^a=y$ on MathOverflow