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 .
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 .
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+{\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{\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 .
(Associated) Laguerre polynomials
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+(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\{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 \}}}
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 . hdl :20.500.11850/133495 .
Möcklin, Egon (1934). Asymptotische Entwicklungen der Laguerreschen Polynome (Thesis). ETH Zurich. doi :10.3929/ethz-a-000092417 . hdl :20.500.11850/133650 .
^ Szegő, Gábor (1975). Orthogonal polynomials . Vol. 4. Providence, Rhode Island: American Mathematical Society. pp. 200–201. ISBN 0-8218-1023-5
Categories :
Plancherel–Rotach asymptotics
Add topic
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
**DISCLAIMER** We are not affiliated with Wikipedia, and Cloudflare.
The information presented on this site is for general informational purposes only and does not constitute medical advice.
You should always have a personal consultation with a healthcare professional before making changes to your diet, medication, or exercise routine.
AI helps with the correspondence in our chat.
We participate in an affiliate program. If you buy something through a link, we may earn a commission 💕
↑
denote the n-th Hermite polynomial. Let 
and 
be positive and fixed, then
and 
and 
and 
complex and bounded
denotes the 
denote the n-th associate Laguerre polynomial. Let 
be arbitrary and real, 
and 
and 
and 
.