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Statements
(
sin
1
2
θ
)
α
+
1
2
(
cos
1
2
θ
)
β
+
1
2
P
n
(
α
,
β
)
(
cos
θ
)
=
π
-
1
2
2
n
+
α
+
β
+
1
B
(
n
+
α
+
1
,
n
+
β
+
1
)
(
∑
m
=
0
M
-
1
f
m
(
θ
)
2
m
(
2
n
+
α
+
β
+
2
)
m
+
O
(
n
-
M
)
)
,
superscript
1
2
𝜃
𝛼
1
2
superscript
1
2
𝜃
𝛽
1
2
Jacobi-polynomial-P
𝛼
𝛽
𝑛
𝜃
superscript
𝜋
1
superscript
2
2
𝑛
𝛼
𝛽
1
Euler-Beta
𝑛
𝛼
1
𝑛
𝛽
1
superscript
subscript
𝑚
0
𝑀
1
subscript
𝑓
𝑚
𝜃
superscript
2
𝑚
Pochhammer
2
𝑛
𝛼
𝛽
2
𝑚
Big-O
superscript
𝑛
𝑀
{\displaystyle{\displaystyle\left(\sin\tfrac{1}{2}\theta\right)^{\alpha+\frac{%
1}{2}}\left(\cos\tfrac{1}{2}\theta\right)^{\beta+\frac{1}{2}}P^{(\alpha,\beta)%
}_{n}\left(\cos\theta\right)=\pi^{-1}2^{2n+\alpha+\beta+1}\mathrm{B}\left(n+%
\alpha+1,n+\beta+1\right)\*\left(\sum_{m=0}^{M-1}\frac{f_{m}(\theta)}{2^{m}{%
\left(2n+\alpha+\beta+2\right)_{m}}}+O\left(n^{-M}\right)\right),}}
O
(
x
)
Big-O
𝑥
{\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
B
(
a
,
b
)
Euler-Beta
𝑎
𝑏
{\displaystyle{\displaystyle\mathrm{B}\left(\NVar{a},\NVar{b}\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}}}}}
π
{\displaystyle{\displaystyle\pi}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
m
𝑚
{\displaystyle{\displaystyle m}}
n
𝑛
{\displaystyle{\displaystyle n}}
f
m
(
θ
)
subscript
𝑓
𝑚
𝜃
{\displaystyle{\displaystyle f_{m}(\theta)}}
Identifiers
