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Statements
L
n
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α
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ν
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=
e
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ν
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x
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2
α
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4
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1
-
x
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1
4
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ξ
1
2
J
α
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ν
ξ
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∑
m
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M
-
1
A
m
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ν
2
m
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ξ
-
1
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J
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+
1
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ν
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∑
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0
M
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B
m
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ν
2
m
+
1
+
ξ
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env
J
α
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ν
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O
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ν
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,
Laguerre-polynomial-L
𝛼
𝑛
𝜈
𝑥
superscript
𝑒
1
2
𝜈
𝑥
superscript
2
𝛼
superscript
𝑥
1
2
𝛼
1
4
superscript
1
𝑥
1
4
superscript
𝜉
1
2
Bessel-J
𝛼
𝜈
𝜉
superscript
subscript
𝑚
0
𝑀
1
subscript
𝐴
𝑚
𝜉
superscript
𝜈
2
𝑚
superscript
𝜉
1
2
Bessel-J
𝛼
1
𝜈
𝜉
superscript
subscript
𝑚
0
𝑀
1
subscript
𝐵
𝑚
𝜉
superscript
𝜈
2
𝑚
1
superscript
𝜉
1
2
envelope-Bessel-J
𝛼
𝜈
𝜉
Big-O
1
superscript
𝜈
2
𝑀
1
{\displaystyle{\displaystyle L^{(\alpha)}_{n}\left(\nu x\right)=\frac{e^{\frac%
{1}{2}\nu x}}{2^{\alpha}x^{\frac{1}{2}\alpha+\frac{1}{4}}(1-x)^{\frac{1}{4}}}%
\left(\xi^{\frac{1}{2}}J_{\alpha}\left(\nu\xi\right)\sum_{m=0}^{M-1}\frac{A_{m%
}(\xi)}{\nu^{2m}}+\xi^{-\frac{1}{2}}J_{\alpha+1}\left(\nu\xi\right)\sum_{m=0}^%
{M-1}\frac{B_{m}(\xi)}{\nu^{2m+1}}+\xi^{\frac{1}{2}}\mathrm{env}\mskip-2.0mu J%
_{\alpha}\left(\nu\xi\right)O\left(\frac{1}{\nu^{2M-1}}\right)\right),}}
J
ν
(
z
)
Bessel-J
𝜈
𝑧
{\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}
O
(
x
)
Big-O
𝑥
{\displaystyle{\displaystyle O\left(\NVar{x}\right)}}
L
n
(
α
)
(
x
)
Laguerre-polynomial-L
𝛼
𝑛
𝑥
{\displaystyle{\displaystyle L^{(\NVar{\alpha})}_{\NVar{n}}\left(\NVar{x}%
\right)}}
env
J
ν
(
x
)
envelope-Bessel-J
𝜈
𝑥
{\displaystyle{\displaystyle\mathrm{env}\mskip-2.0mu J_{\NVar{\nu}}\left(\NVar%
{x}\right)}}
e
{\displaystyle{\displaystyle\mathrm{e}}}
m
𝑚
{\displaystyle{\displaystyle m}}
n
𝑛
{\displaystyle{\displaystyle n}}
ν
𝜈
{\displaystyle{\displaystyle\nu}}
ξ
𝜉
{\displaystyle{\displaystyle\xi}}
A
m
(
ξ
)
subscript
𝐴
𝑚
𝜉
{\displaystyle{\displaystyle A_{m}(\xi)}}
B
m
(
ξ
)
subscript
𝐵
𝑚
𝜉
{\displaystyle{\displaystyle B_{m}(\xi)}}
x
𝑥
{\displaystyle{\displaystyle x}}
