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Statements
Q
ν
μ
(
x
)
=
e
μ
π
i
π
1
/
2
Γ
(
ν
+
μ
+
1
)
(
x
2
-
1
)
μ
/
2
2
ν
+
1
x
ν
+
μ
+
1
𝐅
(
1
2
ν
+
1
2
μ
+
1
,
1
2
ν
+
1
2
μ
+
1
2
;
ν
+
3
2
;
1
x
2
)
,
Legendre-Q-second-kind
𝜇
𝜈
𝑥
superscript
𝑒
𝜇
𝜋
𝑖
superscript
𝜋
1
2
Euler-Gamma
𝜈
𝜇
1
superscript
superscript
𝑥
2
1
𝜇
2
superscript
2
𝜈
1
superscript
𝑥
𝜈
𝜇
1
scaled-hypergeometric-bold-F
1
2
𝜈
1
2
𝜇
1
1
2
𝜈
1
2
𝜇
1
2
𝜈
3
2
1
superscript
𝑥
2
{\displaystyle{\displaystyle Q^{\mu}_{\nu}\left(x\right)=e^{\mu\pi i}\frac{\pi%
^{1/2}\Gamma\left(\nu+\mu+1\right)\left(x^{2}-1\right)^{\mu/2}}{2^{\nu+1}x^{%
\nu+\mu+1}}\mathbf{F}\left(\tfrac{1}{2}\nu+\tfrac{1}{2}\mu+1,\tfrac{1}{2}\nu+%
\tfrac{1}{2}\mu+\tfrac{1}{2};\nu+\tfrac{3}{2};\frac{1}{x^{2}}\right),}}
μ
+
ν
≠
-
1
,
-
2
,
-
3
,
…
𝜇
𝜈
1
2
3
…
{\displaystyle{\displaystyle\mu+\nu\neq-1,-2,-3,\dots}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
Q
ν
μ
(
z
)
Legendre-Q-second-kind
𝜇
𝜈
𝑧
{\displaystyle{\displaystyle Q^{\NVar{\mu}}_{\NVar{\nu}}\left(\NVar{z}\right)}}
π
{\displaystyle{\displaystyle\pi}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
i
imaginary-unit
{\displaystyle{\displaystyle\mathrm{i}}}
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)}}
x
𝑥
{\displaystyle{\displaystyle x}}
μ
𝜇
{\displaystyle{\displaystyle\mu}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
