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Statements
Γ
(
λ
-
μ
)
C
n
(
λ
-
μ
)
(
x
-
1
2
)
x
λ
-
μ
+
1
2
n
=
∫
x
∞
Γ
(
λ
)
C
n
(
λ
)
(
y
-
1
2
)
y
λ
+
1
2
n
(
y
-
x
)
μ
-
1
Γ
(
μ
)
d
y
,
Euler-Gamma
𝜆
𝜇
ultraspherical-Gegenbauer-polynomial
𝜆
𝜇
𝑛
superscript
𝑥
1
2
superscript
𝑥
𝜆
𝜇
1
2
𝑛
superscript
subscript
𝑥
Euler-Gamma
𝜆
ultraspherical-Gegenbauer-polynomial
𝜆
𝑛
superscript
𝑦
1
2
superscript
𝑦
𝜆
1
2
𝑛
superscript
𝑦
𝑥
𝜇
1
Euler-Gamma
𝜇
𝑦
{\displaystyle{\displaystyle\frac{\Gamma\left(\lambda-\mu\right)C^{(\lambda-%
\mu)}_{n}\left(x^{-\frac{1}{2}}\right)}{x^{\lambda-\mu+\frac{1}{2}n}}=\int_{x}%
^{\infty}\frac{\Gamma\left(\lambda\right)C^{(\lambda)}_{n}\left(y^{-\frac{1}{2%
}}\right)}{y^{\lambda+\frac{1}{2}n}}\frac{(y-x)^{\mu-1}}{\Gamma\left(\mu\right%
)}\mathrm{d}y,}}
x
>
0
𝑥
0
{\displaystyle{\displaystyle x>0}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
d
x
𝑥
{\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}
∫
{\displaystyle{\displaystyle\int}}
C
n
(
λ
)
(
x
)
ultraspherical-Gegenbauer-polynomial
𝜆
𝑛
𝑥
{\displaystyle{\displaystyle C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}%
\right)}}
y
𝑦
{\displaystyle{\displaystyle y}}
n
𝑛
{\displaystyle{\displaystyle n}}
x
𝑥
{\displaystyle{\displaystyle x}}
