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Statements
F
2
3
(
a
1
+
2
,
a
2
,
a
3
b
1
,
b
2
;
z
)
a
1
(
a
1
+
1
)
(
1
-
z
)
+
F
2
3
(
a
1
+
1
,
a
2
,
a
3
b
1
,
b
2
;
z
)
a
1
(
b
1
+
b
2
-
3
a
1
-
2
+
z
(
2
a
1
-
a
2
-
a
3
+
1
)
)
+
F
2
3
(
a
1
,
a
2
,
a
3
b
1
,
b
2
;
z
)
(
(
2
a
1
-
b
1
)
(
2
a
1
-
b
2
)
+
a
1
-
a
1
2
-
z
(
a
1
-
a
2
)
(
a
1
-
a
3
)
)
-
F
2
3
(
a
1
-
1
,
a
2
,
a
3
b
1
,
b
2
;
z
)
(
a
1
-
b
1
)
(
a
1
-
b
2
)
=
0
.
Gauss-hypergeometric-pFq
3
2
subscript
𝑎
1
2
subscript
𝑎
2
subscript
𝑎
3
subscript
𝑏
1
subscript
𝑏
2
𝑧
subscript
𝑎
1
subscript
𝑎
1
1
1
𝑧
Gauss-hypergeometric-pFq
3
2
subscript
𝑎
1
1
subscript
𝑎
2
subscript
𝑎
3
subscript
𝑏
1
subscript
𝑏
2
𝑧
subscript
𝑎
1
subscript
𝑏
1
subscript
𝑏
2
3
subscript
𝑎
1
2
𝑧
2
subscript
𝑎
1
subscript
𝑎
2
subscript
𝑎
3
1
Gauss-hypergeometric-pFq
3
2
subscript
𝑎
1
subscript
𝑎
2
subscript
𝑎
3
subscript
𝑏
1
subscript
𝑏
2
𝑧
2
subscript
𝑎
1
subscript
𝑏
1
2
subscript
𝑎
1
subscript
𝑏
2
subscript
𝑎
1
superscript
subscript
𝑎
1
2
𝑧
subscript
𝑎
1
subscript
𝑎
2
subscript
𝑎
1
subscript
𝑎
3
Gauss-hypergeometric-pFq
3
2
subscript
𝑎
1
1
subscript
𝑎
2
subscript
𝑎
3
subscript
𝑏
1
subscript
𝑏
2
𝑧
subscript
𝑎
1
subscript
𝑏
1
subscript
𝑎
1
subscript
𝑏
2
0
{\displaystyle{\displaystyle{{}_{3}F_{2}}\left({a_{1}+2,a_{2},a_{3}\atop b_{1}%
,b_{2}};z\right)a_{1}(a_{1}+1)(1-z)+{{}_{3}F_{2}}\left({a_{1}+1,a_{2},a_{3}%
\atop b_{1},b_{2}};z\right)a_{1}\left(b_{1}+b_{2}-3a_{1}-2+z(2a_{1}-a_{2}-a_{3%
}+1)\right)+{{}_{3}F_{2}}\left({a_{1},a_{2},a_{3}\atop b_{1},b_{2}};z\right)%
\left((2a_{1}-b_{1})(2a_{1}-b_{2})+a_{1}-a_{1}^{2}-z(a_{1}-a_{2})(a_{1}-a_{3})%
\right)-{{}_{3}F_{2}}\left({a_{1}-1,a_{2},a_{3}\atop b_{1},b_{2}};z\right)(a_{%
1}-b_{1})(a_{1}-b_{2})=0.}}
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)}}
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
