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Statements
Γ
(
λ
+
1
2
)
e
z
cos
θ
(
1
2
z
sin
θ
)
1
2
-
λ
J
λ
-
1
2
(
z
sin
θ
)
=
∑
n
=
0
∞
C
n
(
λ
)
(
cos
θ
)
(
2
λ
)
n
z
n
,
Euler-Gamma
𝜆
1
2
superscript
𝑒
𝑧
𝜃
superscript
1
2
𝑧
𝜃
1
2
𝜆
Bessel-J
𝜆
1
2
𝑧
𝜃
superscript
subscript
𝑛
0
ultraspherical-Gegenbauer-polynomial
𝜆
𝑛
𝜃
Pochhammer
2
𝜆
𝑛
superscript
𝑧
𝑛
{\displaystyle{\displaystyle\Gamma\left(\lambda+\tfrac{1}{2}\right)e^{z\cos%
\theta}(\tfrac{1}{2}z\sin\theta)^{\frac{1}{2}-\lambda}J_{\lambda-\frac{1}{2}}%
\left(z\sin\theta\right)=\sum_{n=0}^{\infty}\frac{C^{(\lambda)}_{n}\left(\cos%
\theta\right)}{{\left(2\lambda\right)_{n}}}z^{n},}}
0
≤
θ
≤
π
0
𝜃
𝜋
{\displaystyle{\displaystyle 0\leq\theta\leq\pi}}
J
ν
(
z
)
Bessel-J
𝜈
𝑧
{\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
Γ
(
z
)
Euler-Gamma
𝑧
{\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}
(
a
)
n
Pochhammer
𝑎
𝑛
{\displaystyle{\displaystyle{\left(\NVar{a}\right)_{\NVar{n}}}}}
π
{\displaystyle{\displaystyle\pi}}
cos
z
𝑧
{\displaystyle{\displaystyle\cos\NVar{z}}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
sin
z
𝑧
{\displaystyle{\displaystyle\sin\NVar{z}}}
C
n
(
λ
)
(
x
)
ultraspherical-Gegenbauer-polynomial
𝜆
𝑛
𝑥
{\displaystyle{\displaystyle C^{(\NVar{\lambda})}_{\NVar{n}}\left(\NVar{x}%
\right)}}
z
𝑧
{\displaystyle{\displaystyle z}}
n
𝑛
{\displaystyle{\displaystyle n}}
