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Statements
(
1
+
z
)
-
α
-
β
-
1
F
1
2
(
1
2
(
α
+
β
+
1
)
,
1
2
(
α
+
β
+
2
)
β
+
1
;
2
(
x
+
1
)
z
(
1
+
z
)
2
)
=
∑
n
=
0
∞
(
α
+
β
+
1
)
n
(
β
+
1
)
n
P
n
(
α
,
β
)
(
x
)
z
n
,
superscript
1
𝑧
𝛼
𝛽
1
Gauss-hypergeometric-F-as-2F1
1
2
𝛼
𝛽
1
1
2
𝛼
𝛽
2
𝛽
1
2
𝑥
1
𝑧
superscript
1
𝑧
2
superscript
subscript
𝑛
0
Pochhammer
𝛼
𝛽
1
𝑛
Pochhammer
𝛽
1
𝑛
Jacobi-polynomial-P
𝛼
𝛽
𝑛
𝑥
superscript
𝑧
𝑛
{\displaystyle{\displaystyle(1+z)^{-\alpha-\beta-1}\*{{}_{2}F_{1}}\left({%
\tfrac{1}{2}(\alpha+\beta+1),\tfrac{1}{2}(\alpha+\beta+2)\atop\beta+1};\frac{2%
(x+1)z}{(1+z)^{2}}\right)=\sum_{n=0}^{\infty}\frac{{\left(\alpha+\beta+1\right%
)_{n}}}{{\left(\beta+1\right)_{n}}}P^{(\alpha,\beta)}_{n}\left(x\right)z^{n},}}
|
z
|
<
1
𝑧
1
{\displaystyle{\displaystyle|z|<1}}
F
1
2
(
a
,
b
;
c
;
z
)
Gauss-hypergeometric-F-as-2F1
𝑎
𝑏
𝑐
𝑧
{\displaystyle{\displaystyle{{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};%
\NVar{z}\right)}}
P
n
(
α
,
β
)
(
x
)
Jacobi-polynomial-P
𝛼
𝛽
𝑛
𝑥
{\displaystyle{\displaystyle P^{(\NVar{\alpha},\NVar{\beta})}_{\NVar{n}}\left(%
\NVar{x}\right)}}
(
a
)
n
Pochhammer
𝑎
𝑛
{\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
z
𝑧
{\displaystyle{\displaystyle z}}
n
𝑛
{\displaystyle{\displaystyle n}}
x
𝑥
{\displaystyle{\displaystyle x}}
Identifiers
