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Statements
π―
Ξ½
m
β‘
(
x
)
=
(
-
1
)
m
β’
Ξ
β‘
(
Ξ½
+
m
+
1
)
Ξ
β‘
(
Ξ½
-
m
+
1
)
β’
(
1
-
x
1
+
x
)
m
/
2
β’
π
β‘
(
Ξ½
+
1
,
-
Ξ½
;
m
+
1
;
1
2
-
1
2
β’
x
)
.
Ferrers-Legendre-P-first-kind
π
π
π₯
superscript
1
π
Euler-Gamma
π
π
1
Euler-Gamma
π
π
1
superscript
1
π₯
1
π₯
π
2
scaled-hypergeometric-bold-F
π
1
π
π
1
1
2
1
2
π₯
{\displaystyle{\displaystyle\mathsf{P}^{m}_{\nu}\left(x\right)=(-1)^{m}\frac{%
\Gamma\left(\nu+m+1\right)}{\Gamma\left(\nu-m+1\right)}\left(\frac{1-x}{1+x}%
\right)^{m/2}\mathbf{F}\left(\nu+1,-\nu;m+1;\tfrac{1}{2}-\tfrac{1}{2}x\right).}}
Ξ
β‘
(
z
)
Euler-Gamma
π§
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
π―
Ξ½
ΞΌ
β‘
(
x
)
Ferrers-Legendre-P-first-kind
π
π
π₯
{\displaystyle{\displaystyle\mathsf{P}^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{x}%
\right)}}
π
β‘
(
a
,
b
;
c
;
z
)
scaled-hypergeometric-bold-F
π
π
π
π§
{\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z%
}\right)}}
x
π₯
{\displaystyle{\displaystyle x}}
m
π
{\displaystyle{\displaystyle m}}
Ξ½
π
{\displaystyle{\displaystyle\nu}}
