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Statements
2
sin
(
μ
π
)
π
𝑸
ν
μ
(
x
)
=
(
x
+
1
)
μ
/
2
Γ
(
ν
+
μ
+
1
)
(
x
-
1
)
μ
/
2
𝐅
(
ν
+
1
,
-
ν
;
1
-
μ
;
1
2
-
1
2
x
)
-
(
x
-
1
)
μ
/
2
Γ
(
ν
-
μ
+
1
)
(
x
+
1
)
μ
/
2
𝐅
(
ν
+
1
,
-
ν
;
μ
+
1
;
1
2
-
1
2
x
)
.
2
𝜇
𝜋
𝜋
associated-Legendre-black-Q
𝜇
𝜈
𝑥
superscript
𝑥
1
𝜇
2
Euler-Gamma
𝜈
𝜇
1
superscript
𝑥
1
𝜇
2
scaled-hypergeometric-bold-F
𝜈
1
𝜈
1
𝜇
1
2
1
2
𝑥
superscript
𝑥
1
𝜇
2
Euler-Gamma
𝜈
𝜇
1
superscript
𝑥
1
𝜇
2
scaled-hypergeometric-bold-F
𝜈
1
𝜈
𝜇
1
1
2
1
2
𝑥
{\displaystyle{\displaystyle\frac{2\sin\left(\mu\pi\right)}{\pi}\boldsymbol{Q}%
^{\mu}_{\nu}\left(x\right)=\frac{(x+1)^{\mu/2}}{\Gamma\left(\nu+\mu+1\right)(x%
-1)^{\mu/2}}\mathbf{F}\left(\nu+1,-\nu;1-\mu;\tfrac{1}{2}-\tfrac{1}{2}x\right)%
-\frac{(x-1)^{\mu/2}}{\Gamma\left(\nu-\mu+1\right)(x+1)^{\mu/2}}\mathbf{F}%
\left(\nu+1,-\nu;\mu+1;\tfrac{1}{2}-\tfrac{1}{2}x\right).}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
𝑸
ν
μ
(
z
)
associated-Legendre-black-Q
𝜇
𝜈
𝑧
{\displaystyle{\displaystyle\boldsymbol{Q}^{\NVar{\mu}}_{\NVar{\nu}}\left(%
\NVar{z}\right)}}
π
{\displaystyle{\displaystyle\pi}}
𝐅
(
a
,
b
;
c
;
z
)
scaled-hypergeometric-bold-F
𝑎
𝑏
𝑐
𝑧
{\displaystyle{\displaystyle\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z%
}\right)}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
x
𝑥
{\displaystyle{\displaystyle x}}
μ
𝜇
{\displaystyle{\displaystyle\mu}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
