Fox H-function

{{Short description|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 {{harvs|txt|authorlink=Charles Fox (mathematician)|first=Charles|last=Fox|year=1961}}.

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.

File: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

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 $z$ .{{Cite web |last=Rathie and Ozelim |first=Pushpa Narayan and Luan Carlos de Sena Monteiro |title=On the Relation between Lambert W-Function and Generalized Hypergeometric Functions |url=https://www.researchgate.net/publication/365706509 |access-date=1 March 2023 |website=Researchgate}}

= 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 :{{harv|Srivastava|Manocha|1984|p=50}}

: $H_{p,q}^{\,m,n} \!\left[ z \left| \begin{matrix}
( a_1 , C ) & ( a_2 , C ) & \ldots & ( a_p , C ) \\
( b_1 , C ) & ( b_2 , C ) & \ldots & ( b_q , C ) \end{matrix} \right. \right]
= \frac{1}{C}
G_{p,q}^{\,m,n} \!\left( \left. \begin{matrix} a_1, \dots, a_p \\ b_1, \dots, b_q \end{matrix} \; \right| \, z^{1/C} \right).$

A generalization of the Fox H-function was given by Ram Kishore Saxena.{{Cite book |last1=Mathai |first1=A. M. |url=https://books.google.com/books?id=MvZUAAAAYAAJ |title=Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences |last2=Saxena |first2=R. K. |last3=Saxena |first3=Ram Kishore |date=1973 |publisher=Springer |isbn=978-0-387-06482-6 |language=en}}{{harvtxt|Innayat-Hussain|1987a}} A further generalization of this function, useful in physics and statistics, was provided by A.M. Mathai and Ram Kishore Saxena.{{Cite book |last1=Mathai |first1=A. M. |url=https://books.google.com/books?id=DkLvAAAAMAAJ |title=The H-function with Applications in Statistics and Other Disciplines |last2=Saxena |first2=Rajendra Kumar |date=1978 |publisher=Wiley |isbn=978-0-470-26380-8 |language=en}}{{harvtxt|Rathie|1997}}

References

{{Citation | last1= Fox | first1= Charles | title= The G and H functions as symmetrical Fourier kernels | jstor= 1993339 | mr= 0131578 | year= 1961 | journal= Transactions of the American Mathematical Society | issn= 0002-9947 | volume= 98 | issue= 3 | pages= 395–429 | doi=10.2307/1993339}}
{{Citation | last= Innayat-Hussain | first=AA | title= New properties of hypergeometric series derivable from Feynman integrals. I: Transformation and reduction formulae | journal= J. Phys. A: Math. Gen. | volume= 20 | year= 1987a | issue=13 | pages= 4109–4117 | doi= 10.1088/0305-4470/20/13/019 | bibcode=1987JPhA...20.4109I }}
{{Citation | last= Innayat-Hussain | first=AA | title= New properties of hypergeometric series derivable from Feynman integrals. II: A generalization of the H-function | journal= J. Phys. A: Math. Gen. | volume= 20 | year= 1987b | issue=13 | pages= 4119–4128 | doi= 10.1088/0305-4470/20/13/020 | bibcode=1987JPhA...20.4119I }}
{{Citation

| last1= Kilbas | first1= Anatoly A.

| title=H-Transforms: Theory and Applications

| publisher= CRC Press

| isbn= 978-0415299169

| year= 2004 }}

{{Citation | last1= Mathai | first1= A. M. | last2=Saxena | first2=Ram Kishore | title= The H-function with applications in statistics and other disciplines | publisher= Halsted Press [John Wiley & Sons], New York-London-Sidney | isbn= 978-0-470-26380-8 | mr=513025 | year= 1978 }}
{{Citation | last1= Mathai | first1= A. M. | last2= Saxena | first2= Ram Kishore | last3= Haubold | first3= Hans J. | title= The H-function | publisher= Springer-Verlag | location= Berlin, New York | isbn= 978-1-4419-0915-2 | mr= 2562766 | year= 2010 }}
{{Citation | last= Rathie | first= Arjun K. | title= A new generalization of generalized hypergeometric function | journal= Le Matematiche | volume= LII | year= 1997 | pages= 297–310 }}.
{{Citation | last1= Srivastava | first1= H. M. | last2= Gupta | first2= K. C. | last3= Goyal | first3= S. P. | title= The H-functions of one and two variables | publisher= South Asian Publishers Pvt. Ltd. | location= New Delhi | mr= 691138 | year= 1982 }}
{{cite book | last1= Srivastava | first1= H. M. | last2= Manocha | first2= H. L. | title= A treatise on generating functions | year= 1984 | publisher= E. Horwood | isbn= 0-470-20010-3 }}

External links

[https://gitlab.com/RZ-FZJ/hypergeom hypergeom] on GitLab
[https://mathoverflow.net/questions/407760/is-there-a-specific-named-function-that-is-the-inverse-of-xxa-for-x-real/407777#407777 Use in solving $x+x^a=y$ ] on MathOverflow

Category:Hypergeometric functions

Category:Special functions