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Statements
P
ν
μ
(
x
)
=
2
μ
Γ
(
1
-
2
μ
)
Γ
(
ν
+
μ
+
1
)
Γ
(
ν
-
μ
+
1
)
Γ
(
1
-
μ
)
(
x
2
-
1
)
μ
/
2
C
ν
+
μ
(
1
2
-
μ
)
(
x
)
.
Legendre-P-first-kind
𝜇
𝜈
𝑥
superscript
2
𝜇
Euler-Gamma
1
2
𝜇
Euler-Gamma
𝜈
𝜇
1
Euler-Gamma
𝜈
𝜇
1
Euler-Gamma
1
𝜇
superscript
superscript
𝑥
2
1
𝜇
2
ultraspherical-Gegenbauer-polynomial
1
2
𝜇
𝜈
𝜇
𝑥
{\displaystyle{\displaystyle P^{\mu}_{\nu}\left(x\right)=\frac{2^{\mu}\Gamma%
\left(1-2\mu\right)\Gamma\left(\nu+\mu+1\right)}{\Gamma\left(\nu-\mu+1\right)%
\Gamma\left(1-\mu\right)\left(x^{2}-1\right)^{\mu/2}}C^{(\frac{1}{2}-\mu)}_{%
\nu+\mu}\left(x\right).}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
P
ν
μ
(
z
)
Legendre-P-first-kind
𝜇
𝜈
𝑧
{\displaystyle{\displaystyle P^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
C
n
(
λ
)
(
x
)
ultraspherical-Gegenbauer-polynomial
𝜆
𝑛
𝑥
{\displaystyle{\displaystyle C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}%
\right)}}
x
𝑥
{\displaystyle{\displaystyle x}}
μ
𝜇
{\displaystyle{\displaystyle\mu}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
