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Statements
F
1
2
(
a
,
b
c
;
𝐓
)
=
1
B
m
(
a
,
c
-
a
)
∫
𝟎
<
𝐗
<
𝐈
|
𝐗
|
a
-
1
2
(
m
+
1
)
|
𝐈
-
𝐗
|
c
-
a
-
1
2
(
m
+
1
)
|
𝐈
-
𝐓𝐗
|
-
b
d
𝐗
,
Gauss-hypergeometric-of-matrix-pFq
2
1
𝑎
𝑏
𝑐
𝐓
1
multivariate-Euler-Beta
𝑚
𝑎
𝑐
𝑎
subscript
0
𝐗
𝐈
superscript
𝐗
𝑎
1
2
𝑚
1
superscript
𝐈
𝐗
𝑐
𝑎
1
2
𝑚
1
superscript
𝐈
𝐓𝐗
𝑏
𝐗
{\displaystyle{\displaystyle{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=%
\frac{1}{\mathrm{B}_{m}\left(a,c-a\right)}\int\limits_{\boldsymbol{{0}}<%
\mathbf{X}<\mathbf{I}}|\mathbf{X}|^{a-\frac{1}{2}(m+1)}\*{|\mathbf{I}-\mathbf{%
X}|}^{c-a-\frac{1}{2}(m+1)}{|\mathbf{I}-\mathbf{T}\mathbf{X}|}^{-b}\mathrm{d}%
\mathbf{X},}}
ℜ
(
a
)
,
ℜ
(
c
-
a
)
>
1
2
(
m
-
1
)
𝑎
𝑐
𝑎
1
2
𝑚
1
{\displaystyle{\displaystyle\Re(a),\Re(c-a)>\frac{1}{2}(m-1)}}
ℜ
(
a
)
,
ℜ
(
c
-
a
)
>
1
2
(
m
-
1
)
𝑎
𝑐
𝑎
1
2
𝑚
1
{\displaystyle{\displaystyle\Re(a),\Re(c-a)>\frac{1}{2}(m-1)}}
𝟎
<
𝐓
<
𝐈
0
𝐓
𝐈
{\displaystyle{\displaystyle\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}}}
F
q
p
(
a
1
,
…
,
a
p
;
b
1
,
…
,
b
q
;
𝐓
)
Gauss-hypergeometric-of-matrix-pFq
𝑝
𝑞
subscript
𝑎
1
…
subscript
𝑎
𝑝
subscript
𝑏
1
…
subscript
𝑏
𝑞
𝐓
{\displaystyle{\displaystyle{{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots%
,a_{p}};\NVar{b_{1},\dots,b_{q}};\NVar{\mathbf{T}}\right)}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
∫
{\displaystyle{\displaystyle\int}}
B
m
(
a
,
b
)
multivariate-Euler-Beta
𝑚
𝑎
𝑏
{\displaystyle{\displaystyle\mathrm{B}_{\NVar{m}}\left(\NVar{a},\NVar{b}\right%
)}}
ℜ
absent
{\displaystyle{\displaystyle\Re}}
a
𝑎
{\displaystyle{\displaystyle a}}
𝐓
𝐓
{\displaystyle{\displaystyle\mathbf{T}}}
𝐗
𝐗
{\displaystyle{\displaystyle\mathbf{X}}}
b
𝑏
{\displaystyle{\displaystyle b}}
c
𝑐
{\displaystyle{\displaystyle c}}
m
𝑚
{\displaystyle{\displaystyle m}}
