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