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Statements
𝖯
n
m
(
x
)
=
(
1
2
)
m
(
-
2
)
m
(
1
-
x
2
)
1
2
m
C
n
-
m
(
m
+
1
2
)
(
x
)
=
(
n
+
1
)
m
(
-
2
)
-
m
(
1
-
x
2
)
1
2
m
P
n
-
m
(
m
,
m
)
(
x
)
,
Ferrers-Legendre-P-first-kind
𝑚
𝑛
𝑥
Pochhammer
1
2
𝑚
superscript
2
𝑚
superscript
1
superscript
𝑥
2
1
2
𝑚
ultraspherical-Gegenbauer-polynomial
𝑚
1
2
𝑛
𝑚
𝑥
Pochhammer
𝑛
1
𝑚
superscript
2
𝑚
superscript
1
superscript
𝑥
2
1
2
𝑚
Jacobi-polynomial-P
𝑚
𝑚
𝑛
𝑚
𝑥
{\displaystyle{\displaystyle\mathsf{P}^{m}_{n}\left(x\right)={\left(\tfrac{1}{%
2}\right)_{m}}(-2)^{m}(1-x^{2})^{\frac{1}{2}m}C^{(m+\frac{1}{2})}_{n-m}\left(x%
\right)={\left(n+1\right)_{m}}(-2)^{-m}(1-x^{2})^{\frac{1}{2}m}P^{(m,m)}_{n-m}%
\left(x\right),}}
0
≤
m
≤
n
0
𝑚
𝑛
{\displaystyle{\displaystyle 0\leq m\leq n}}
𝖯
ν
μ
(
x
)
Ferrers-Legendre-P-first-kind
𝜇
𝜈
𝑥
{\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}%
\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}}}}}
C
n
(
λ
)
(
x
)
ultraspherical-Gegenbauer-polynomial
𝜆
𝑛
𝑥
{\displaystyle{\displaystyle C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}%
\right)}}
m
𝑚
{\displaystyle{\displaystyle m}}
n
𝑛
{\displaystyle{\displaystyle n}}
x
𝑥
{\displaystyle{\displaystyle x}}
