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 Charles Fox (1961 ).
It is defined by a Mellin–Barnes integral
H
p
,
q
m
,
n
[
z
|
(
a
1
,
A
1
)
(
a
2
,
A
2
)
…
(
a
p
,
A
p
)
(
b
1
,
B
1
)
(
b
2
,
B
2
)
…
(
b
q
,
B
q
)
]
=
1
2
π
i
∫
L
∏
j
=
1
m
Γ
(
b
j
+
B
j
s
)
∏
j
=
1
n
Γ
(
1
−
a
j
−
A
j
s
)
∏
j
=
m
+
1
q
Γ
(
1
−
b
j
−
B
j
s
)
∏
j
=
n
+
1
p
Γ
(
a
j
+
A
j
s
)
z
−
s
d
s
,
{\displaystyle 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_{j}s)\,\prod _{j=1}^{n}\Gamma (1-a_{j}-A_{j}s)}{\prod _{j=m+1}^{q}\Gamma (1-b_{j}-B_{j}s)\,\prod _{j=n+1}^{p}\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) ![{\displaystyle 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_{j}s)\,\prod _{j=1}^{n}\Gamma (1-a_{j}-A_{j}s)}{\prod _{j=m+1}^{q}\Gamma (1-b_{j}-B_{j}s)\,\prod _{j=n+1}^{p}\Gamma (a_{j}+A_{j}s)}}z^{-s}\,ds,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1e0af72e03a0f3221f7f5241446802e999ff5b4)
,...,(a_n,%CE%B1_n)),((a_n%2B1,%CE%B1_n%2B1),...,(a_p,%CE%B1_p)),(((b_1,%CE%B2_1),...,(b_m,%CE%B2_m)),in_((b_m%2B1,%CE%B2_m%2B1),...,(b_q,%CE%B2_q))),z)_with_H(((),()),(((-1,%C2%BD)),()),z).svg)