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Statements
F
q
+
1
p
+
1
(
a
0
,
…
,
a
p
b
0
,
…
,
b
q
;
z
)
=
Γ
(
b
0
)
Γ
(
a
0
)
Γ
(
b
0
-
a
0
)
∫
0
1
t
a
0
-
1
(
1
-
t
)
b
0
-
a
0
-
1
F
q
p
(
a
1
,
…
,
a
p
b
1
,
…
,
b
q
;
z
t
)
d
t
,
Gauss-hypergeometric-pFq
𝑝
1
𝑞
1
subscript
𝑎
0
…
subscript
𝑎
𝑝
subscript
𝑏
0
…
subscript
𝑏
𝑞
𝑧
Euler-Gamma
subscript
𝑏
0
Euler-Gamma
subscript
𝑎
0
Euler-Gamma
subscript
𝑏
0
subscript
𝑎
0
superscript
subscript
0
1
superscript
𝑡
subscript
𝑎
0
1
superscript
1
𝑡
subscript
𝑏
0
subscript
𝑎
0
1
Gauss-hypergeometric-pFq
𝑝
𝑞
subscript
𝑎
1
…
subscript
𝑎
𝑝
subscript
𝑏
1
…
subscript
𝑏
𝑞
𝑧
𝑡
𝑡
{\displaystyle{\displaystyle{{}_{p+1}F_{q+1}}\left({a_{0},\dots,a_{p}\atop b_{%
0},\dots,b_{q}};z\right)=\frac{\Gamma\left(b_{0}\right)}{\Gamma\left(a_{0}%
\right)\Gamma\left(b_{0}-a_{0}\right)}\int_{0}^{1}t^{a_{0}-1}(1-t)^{b_{0}-a_{0%
}-1}{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};zt\right)%
\mathrm{d}t,}}
ℜ
b
0
>
ℜ
a
0
>
0
subscript
𝑏
0
subscript
𝑎
0
0
{\displaystyle{\displaystyle\Re b_{0}>\Re a_{0}>0}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
F
q
p
(
a
1
,
…
,
a
p
;
b
1
,
…
,
b
q
;
z
)
Gauss-hypergeometric-pFq
𝑝
𝑞
subscript
𝑎
1
…
subscript
𝑎
𝑝
subscript
𝑏
1
…
subscript
𝑏
𝑞
𝑧
{\displaystyle{\displaystyle{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots%
,a_{p}};\NVar{b_{1},\dots,b_{q}};\NVar{z}\right)}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
∫
{\displaystyle{\displaystyle\int}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
p
𝑝
{\displaystyle{\displaystyle p}}
q
𝑞
{\displaystyle{\displaystyle q}}
z
𝑧
{\displaystyle{\displaystyle z}}
a
,
a
1
,
…
,
a
p
𝑎
subscript
𝑎
1
…
subscript
𝑎
𝑝
{\displaystyle{\displaystyle a,a_{1},\ldots,a_{p}}}
b
,
b
1
,
…
,
b
q
𝑏
subscript
𝑏
1
…
subscript
𝑏
𝑞
{\displaystyle{\displaystyle b,b_{1},\ldots,b_{q}}}
Identifiers
