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Asymptotic values of Hermite or Laguerre polynomials
The Plancherel–Rotach asymptotics are asymptotic results for orthogonal polynomials . They are named after the Swiss mathematicians Michel Plancherel and his PhD student Walter Rotach , who first derived the asymptotics for the Hermite polynomial and Laguerre polynomial . Nowadays asymptotic expansions of this kind for orthogonal polynomials are referred to as Plancherel–Rotach asymptotics or of Plancherel–Rotach type .[ 1]
The case for the associated Laguerre polynomial was derived by the Swiss mathematician Egon Möcklin , another PhD student of Plancherel and George Pólya at ETH Zurich .[ 2]
Hermite polynomials
Let
H
n
(
x
)
{\displaystyle H_{n}(x)}
ϵ
{\displaystyle \epsilon }
ω
{\displaystyle \omega }
for
x
=
(
2
n
+
1
)
1
/
2
cos
φ
{\displaystyle x=(2n+1)^{1/2}\cos \varphi }
ϵ
≤
φ
≤
π
−
ϵ
{\displaystyle \epsilon \leq \varphi \leq \pi -\epsilon }
e
−
x
2
/
2
H
n
(
x
)
=
2
n
/
2
+
1
/
4
(
n
!
)
1
/
2
(
π
n
)
−
1
/
4
(
sin
φ
)
−
1
/
2
{
sin
[
(
n
2
+
1
4
)
(
sin
2
φ
−
2
φ
)
+
3
π
4
]
+
O
(
n
−
1
)
}
{\displaystyle e^{-x^{2}/2}H_{n}(x)=2^{n/2+1/4}(n!)^{1/2}(\pi n)^{-1/4}(\sin \varphi )^{-1/2}{\bigg \{}\sin \left[\left({\tfrac {n}{2}}+{\tfrac {1}{4}}\right)(\sin 2\varphi -2\varphi )+3{\tfrac {\pi }{4}}\right]+{\mathcal {O}}(n^{-1}){\bigg \}}}
for
x
=
(
2
n
+
1
)
1
/
2
cosh
φ
{\displaystyle x=(2n+1)^{1/2}\cosh \varphi }
ϵ
≤
φ
≤
ω
{\displaystyle \epsilon \leq \varphi \leq \omega }
e
−
x
2
/
2
H
n
(
x
)
=
2
n
/
2
−
3
/
4
(
n
!
)
1
/
2
(
π
n
)
−
1
/
4
(
sinh
φ
)
−
1
/
2
exp
[
(
n
2
+
1
4
)
(
2
φ
−
sinh
2
φ
)
]
{
1
+
O
(
n
−
1
)
}
{\displaystyle e^{-x^{2}/2}H_{n}(x)=2^{n/2-3/4}(n!)^{1/2}(\pi n)^{-1/4}(\sinh \varphi )^{-1/2}\exp \left[\left({\tfrac {n}{2}}+{\tfrac {1}{4}}\right)(2\varphi -\sinh 2\varphi )\right]{\big \{}1+{\mathcal {O}}(n^{-1}){\big \}}}
for
x
=
(
2
n
+
1
)
1
/
2
−
2
−
1
/
2
3
−
1
/
3
n
−
1
/
6
t
{\displaystyle x=(2n+1)^{1/2}-2^{-1/2}3^{-1/3}n^{-1/6}t}
t
{\displaystyle t}
e
−
x
2
/
2
H
n
(
x
)
=
3
1
/
3
π
−
3
/
4
2
n
/
2
+
1
/
4
(
n
!
)
1
/
2
n
−
1
/
12
{
Ai
(
t
)
+
O
(
n
−
2
/
3
)
}
{\displaystyle e^{-x^{2}/2}H_{n}(x)=3^{1/3}\pi ^{-3/4}2^{n/2+1/4}(n!)^{1/2}n^{-1/12}{\bigg \{}\operatorname {Ai} (t)+{\mathcal {O}}\left(n^{-{2/3}}\right){\bigg \}}}
where
Ai
{\displaystyle \operatorname {Ai} }
Airy function .[ 3]
Let
L
n
(
α
)
(
x
)
{\displaystyle L_{n}^{(\alpha )}(x)}
α
{\displaystyle \alpha }
ϵ
{\displaystyle \epsilon }
ω
{\displaystyle \omega }
for
x
=
(
4
n
+
2
α
+
2
)
cos
2
φ
{\displaystyle x=(4n+2\alpha +2)\cos ^{2}\varphi }
ϵ
≤
φ
≤
π
2
−
ϵ
n
−
1
/
2
{\displaystyle \epsilon \leq \varphi \leq {\tfrac {\pi }{2}}-\epsilon n^{-1/2}}
e
−
x
/
2
L
n
(
α
)
(
x
)
=
(
−
1
)
n
(
π
sin
φ
)
−
1
/
2
x
−
α
/
2
−
1
/
4
n
α
/
2
−
1
/
4
{
sin
[
(
n
+
α
+
1
2
)
(
sin
2
φ
−
2
φ
)
+
3
π
/
4
]
+
(
n
x
)
−
1
/
2
O
(
1
)
}
{\displaystyle e^{-x/2}L_{n}^{(\alpha )}(x)=(-1)^{n}(\pi \sin \varphi )^{-1/2}x^{-\alpha /2-1/4}n^{\alpha /2-1/4}{\big \{}\sin \left[\left(n+{\tfrac {\alpha +1}{2}}\right)(\sin 2\varphi -2\varphi )+3\pi /4\right]+(nx)^{-1/2}{\mathcal {O}}(1){\big \}}}
for
x
=
(
4
n
+
2
α
+
2
)
cosh
2
φ
{\displaystyle x=(4n+2\alpha +2)\cosh ^{2}\varphi }
ϵ
≤
φ
≤
ω
{\displaystyle \epsilon \leq \varphi \leq \omega }
e
−
x
/
2
L
n
(
α
)
(
x
)
=
1
2
(
−
1
)
n
(
π
sinh
φ
)
−
1
/
2
x
−
α
/
2
−
1
/
4
n
α
/
2
−
1
/
4
exp
[
(
n
+
α
+
1
2
)
(
2
φ
−
sinh
2
φ
)
]
{
1
+
O
(
n
−
1
)
}
{\displaystyle e^{-x/2}L_{n}^{(\alpha )}(x)={\tfrac {1}{2}}(-1)^{n}(\pi \sinh \varphi )^{-1/2}x^{-\alpha /2-1/4}n^{\alpha /2-1/4}\exp \left[\left(n+{\tfrac {\alpha +1}{2}}\right)(2\varphi -\sinh 2\varphi )\right]\{1+{\mathcal {O}}\left(n^{-1}\right)\}}
for
x
=
4
n
+
2
α
+
2
−
2
(
2
n
/
3
)
1
/
3
t
{\displaystyle x=4n+2\alpha +2-2(2n/3)^{1/3}t}
t
{\displaystyle t}
e
−
x
/
2
L
n
(
α
)
(
x
)
=
(
−
1
)
n
π
−
1
2
−
α
−
1
/
3
3
1
/
3
n
−
1
/
3
{
Ai
(
t
)
+
O
(
n
−
2
/
3
)
}
{\displaystyle e^{-x/2}L_{n}^{(\alpha )}(x)=(-1)^{n}\pi ^{-1}2^{-\alpha -1/3}3^{1/3}n^{-1/3}{\bigg \{}\operatorname {Ai} (t)+{\mathcal {O}}\left(n^{-2/3}\right){\bigg \}}}
[ 3]
Literature
Szegő, Gábor (1975). Orthogonal polynomials . Vol. 4. Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-1023-5
References
^ Rotach, Walter (1925). Reihenentwicklungen einer willkürlichen Funktion nach Hermite'schen und Laguerre'schen Polynomen (Thesis). ETH Zurich. doi :10.3929/ethz-a-000092029 . ^ Möcklin, Egon (1934). Asymptotische Entwicklungen der Laguerreschen Polynome (Thesis). ETH Zurich. doi :10.3929/ethz-a-000092417 . ^ a b Szegő, Gábor (1975). Orthogonal polynomials . Vol. 4. Providence, Rhode Island: American Mathematical Society. pp. 200– 201. ISBN 0-8218-1023-5 
![{\displaystyle e^{-x^{2}/2}H_{n}(x)=2^{n/2+1/4}(n!)^{1/2}(\pi n)^{-1/4}(\sin \varphi )^{-1/2}{\bigg \{}\sin \left[\left({\tfrac {n}{2}}+{\tfrac {1}{4}}\right)(\sin 2\varphi -2\varphi )+3{\tfrac {\pi }{4}}\right]+{\mathcal {O}}(n^{-1}){\bigg \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5327e7c814aefd0efebaa32294bf2415295a64)
![{\displaystyle e^{-x^{2}/2}H_{n}(x)=2^{n/2-3/4}(n!)^{1/2}(\pi n)^{-1/4}(\sinh \varphi )^{-1/2}\exp \left[\left({\tfrac {n}{2}}+{\tfrac {1}{4}}\right)(2\varphi -\sinh 2\varphi )\right]{\big \{}1+{\mathcal {O}}(n^{-1}){\big \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee96365402da3c116bbee8c83a71ecc69f83bc4d)
![{\displaystyle e^{-x/2}L_{n}^{(\alpha )}(x)=(-1)^{n}(\pi \sin \varphi )^{-1/2}x^{-\alpha /2-1/4}n^{\alpha /2-1/4}{\big \{}\sin \left[\left(n+{\tfrac {\alpha +1}{2}}\right)(\sin 2\varphi -2\varphi )+3\pi /4\right]+(nx)^{-1/2}{\mathcal {O}}(1){\big \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6119fe41d5a8158dcb3367e17825744761b19b35)
![{\displaystyle e^{-x/2}L_{n}^{(\alpha )}(x)={\tfrac {1}{2}}(-1)^{n}(\pi \sinh \varphi )^{-1/2}x^{-\alpha /2-1/4}n^{\alpha /2-1/4}\exp \left[\left(n+{\tfrac {\alpha +1}{2}}\right)(2\varphi -\sinh 2\varphi )\right]\{1+{\mathcal {O}}\left(n^{-1}\right)\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22b07b923bd08a7f942ce6fec0e9faf5456ab485)
