Développements asymptotiques de Plancherel-Rotach
Polynôme d'Hermite Soient
ϵ
{\displaystyle \epsilon }
ω
{\displaystyle \omega }
pour
x
=
(
2
n
+
1
)
1
/
2
cos
?/mo>
φ
,
ϵ
?/mo>
φ
?/mo>
π
?/mo>
ϵ
{\displaystyle x=(2n+1)^{1/2}\cos \varphi ,\ \epsilon \leq \varphi \leq \pi -\epsilon }
e
?/mo>
x
2
/
2
H
n
(
x
)
=
2
n
/
2
+
1
/
4
(
n
!
)
1
/
2
(
π
n
)
?/mo>
1
/
4
(
sin
?/mo>
φ
)
?/mo>
1
/
2
{
sin
?/mo>
[
(
n
2
+
1
4
)
(
sin
?/mo>
2
φ
?/mo>
2
φ
)
+
3
π
4
]
+
O
(
n
?/mo>
1
)
}
{\displaystyle {\rm {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 \}}}
pour
x
=
(
2
n
+
1
)
1
/
2
cosh
?/mo>
φ
,
ϵ
?/mo>
φ
?/mo>
ω
{\displaystyle x=(2n+1)^{1/2}\cosh \varphi ,\ \epsilon \leq \varphi \leq \omega }
e
?/mo>
x
2
/
2
H
n
(
x
)
=
2
n
/
2
?/mo>
3
/
4
(
n
!
)
1
/
2
(
π
n
)
?/mo>
1
/
4
(
sinh
?/mo>
φ
)
?/mo>
1
/
2
exp
?/mo>
[
(
n
2
+
1
4
)
(
2
φ
?/mo>
sinh
?/mo>
2
φ
)
]
{
1
+
O
(
n
?/mo>
1
)
}
{\displaystyle {\rm {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 \}}}
pour
x
=
(
2
n
+
1
)
1
/
2
?/mo>
2
?/mo>
1
/
2
3
?/mo>
1
/
3
n
?/mo>
1
/
6
t
{\displaystyle x=(2n+1)^{1/2}-2^{-1/2}3^{-1/3}n^{-1/6}t}
t
{\displaystyle t}
e
?/mo>
x
2
/
2
H
n
(
x
)
=
3
1
/
3
π
?/mo>
3
/
4
2
n
/
2
+
1
/
4
(
n
!
)
1
/
2
n
?/mo>
1
/
12
{
Ai
?/mo>
(
t
)
+
O
(
n
?/mo>
2
/
3
)
}
{\displaystyle {\rm {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 \}}}
où
Ai
{\displaystyle \operatorname {Ai} }
fonction d'Airy .
Polynôme de Laguerre Soit
α
{\displaystyle \alpha }
ϵ
{\displaystyle \epsilon }
ω
{\displaystyle \omega }
pour
x
=
(
4
n
+
2
α
+
2
)
cos
2
?/mo>
φ
,
ϵ
?/mo>
φ
?/mo>
π
2
?/mo>
ϵ
n
?/mo>
1
/
2
{\displaystyle x=(4n+2\alpha +2)\cos ^{2}\varphi ,\ \epsilon \leq \varphi \leq {\tfrac {\pi }{2}}-\epsilon n^{-1/2}}
e
?/mo>
x
/
2
L
n
(
α
)
(
x
)
=
(
?/mo>
1
)
n
(
π
sin
?/mo>
φ
)
?/mo>
1
/
2
x
?/mo>
α
/
2
?/mo>
1
/
4
n
α
/
2
?/mo>
1
/
4
?/mo>
{
sin
?/mo>
[
(
n
+
α
+
1
2
)
(
sin
?/mo>
2
φ
?/mo>
2
φ
)
+
3
π
/
4
]
+
(
n
x
)
?/mo>
1
/
2
O
(
1
)
}
{\displaystyle {\rm {e}}^{-x/2}L_{n}^{(\alpha )}(x)=(-1)^{n}(\pi \sin \varphi )^{-1/2}x^{-\alpha /2-1/4}n^{\alpha /2-1/4}\cdot {\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 \}}}
pour
x
=
(
4
n
+
2
α
+
2
)
cosh
2
?/mo>
φ
,
ϵ
?/mo>
φ
?/mo>
ω
{\displaystyle x=(4n+2\alpha +2)\cosh ^{2}\varphi ,\ \epsilon \leq \varphi \leq \omega }
e
?/mo>
x
/
2
L
n
(
α
)
(
x
)
=
1
2
(
?/mo>
1
)
n
(
π
sinh
?/mo>
φ
)
?/mo>
1
/
2
x
?/mo>
α
/
2
?/mo>
1
/
4
n
α
/
2
?/mo>
1
/
4
?/mo>
exp
?/mo>
[
(
n
+
α
+
1
2
)
(
2
φ
?/mo>
sinh
?/mo>
2
φ
)
]
{
1
+
O
(
n
?/mo>
1
)
}
{\displaystyle {\rm {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}\cdot \exp \left[\left(n+{\tfrac {\alpha +1}{2}}\right)(2\varphi -\sinh 2\varphi )\right]\{1+{\mathcal {O}}\left(n^{-1}\right)\}}
pour
x
=
4
n
+
2
α
+
2
?/mo>
2
(
2
n
/
3
)
1
/
3
t
{\displaystyle x=4n+2\alpha +2-2(2n/3)^{1/3}t}
t
{\displaystyle t}
e
?/mo>
x
/
2
L
n
(
α
)
(
x
)
=
(
?/mo>
1
)
n
π
?/mo>
1
2
?/mo>
α
?/mo>
1
/
3
3
1
/
3
n
?/mo>
1
/
3
{
Ai
?/mo>
(
t
)
+
O
(
n
?/mo>
2
/
3
)
}
{\displaystyle {\rm {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 \}}}
Notes et références Egon Möcklin, Asymptotische Entwicklungen der Laguerreschen Polynome , 1934 (DOI 10.3929/ethz-a-000092417 G. Szegő , Orthogonal polynomials , Providence, Rhode Island, American Mathematical Society, 1975 , 4e éd. (ISBN 0-8218-1023-5 ) , p. 200?01
Bibliographie (en) Thorsten Neuschel, « Plancherel-Rotach formulae for average characteristic polynomials of products of Ginibre random matrices and the Fuss-Catalan distribution » , Random Matrices: Theory and Applications vol. 3, no 1,?/span> 2014 (DOI 10.48550/arXiv.1311.0365 (en) Xiang-Sheng Wang, « Plancherel–Rotach asymptotics of second-order difference equations with linear coefficients » , Journal of Approximation Theory vol. 188,?/span> 2014 , p. 1--18 (DOI 10.1016/j.jat.2014.08.003
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![{\displaystyle {\rm {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://img.franco.wiki/i/38c40863e2005349495ce1406ee69fbd777a7339.svg)
![{\displaystyle {\rm {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://img.franco.wiki/i/18323bd7d4f36586f04425e42dd6524274a88473.svg)
![{\displaystyle {\rm {e}}^{-x/2}L_{n}^{(\alpha )}(x)=(-1)^{n}(\pi \sin \varphi )^{-1/2}x^{-\alpha /2-1/4}n^{\alpha /2-1/4}\cdot {\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://img.franco.wiki/i/0ec5c5cd259ae639e773c4cac57e4e9e01dce37d.svg)
![{\displaystyle {\rm {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}\cdot \exp \left[\left(n+{\tfrac {\alpha +1}{2}}\right)(2\varphi -\sinh 2\varphi )\right]\{1+{\mathcal {O}}\left(n^{-1}\right)\}}](https://img.franco.wiki/i/9e1bac44a023185cf4115c7303203de9832d30f4.svg)
