Plancherel–Rotach asymptotics

{{Short description|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.{{cite thesis|title=Reihenentwicklungen einer willkürlichen Funktion nach Hermite'schen und Laguerre'schen Polynomen|first=Walter|last=Rotach|date=1925|doi=10.3929/ethz-a-000092029|publisher=ETH Zurich|hdl=20.500.11850/133495 }}

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.{{cite thesis|first=Egon|last=Möcklin|title=Asymptotische Entwicklungen der Laguerreschen Polynome|date=1934|publisher=ETH Zurich|doi=10.3929/ethz-a-000092417|hdl=20.500.11850/133650 }}

Hermite polynomials

Let $H_n(x)$ denote the n-th Hermite polynomial. Let $\epsilon$ and $\omega$ be positive and fixed, then

for $x =(2n+1)^{1/2}\cos \varphi$ and $\epsilon \leq \varphi \leq \pi -\epsilon$

:: $e^{-x^2/2}H_n(x)
=2^{n/2+1/4}(n!)^{1/2}(\pi n)^{-1/4}(\sin \varphi)^{-1/2}
\bigg\{\sin\left[\left(\tfrac{n}{2}+\tfrac{1}{4}\right)(\sin 2\varphi-2\varphi)+3\tfrac{ \pi}{4}\right]+\mathcal{O}(n^{-1})\bigg\}$

for $x =(2n+1)^{1/2}\cosh \varphi$ and $\epsilon \leq \varphi \leq \omega$

:: $e^{-x^2/2}H_n(x)
=2^{n/2-3/4}(n!)^{1/2}(\pi n)^{-1/4}(\sinh \varphi)^{-1/2}
\exp\left[\left(\tfrac{n}{2}+\tfrac{1}{4}\right)(2\varphi-\sinh 2\varphi)\right]
\big\{1+\mathcal{O}(n^{-1})\big\}$

for $x =(2n+1)^{1/2}-2^{-1/2}3^{-1/3}n^{-1/6}t$ and $t$ complex and bounded

:: $e^{-x^2/2}H_n(x)
=3^{1/3}\pi^{-3/4}2^{n/2+1/4}(n!)^{1/2}n^{-1/12}
\bigg\{A(t)+\mathcal{O}\left(n^{-{2/3}}\right)\bigg\}$

where $A(t) = \pi \operatorname{Ai}(-3^{-1/3}t)$ and $\operatorname{Ai}$ denotes the Airy function.{{cite book|first1=Gábor|last1=Szegő|title=Orthogonal polynomials|volume=4|publisher=American Mathematical Society|place=Providence, Rhode Island|pages=200–201|date=1975|isbn=0-8218-1023-5}}

(Associated) Laguerre polynomials

Let $L^{(\alpha )}_n(x)$ denote the n-th associate Laguerre polynomial. Let $\alpha$ be arbitrary and real, $\epsilon$ and $\omega$ be positive and fixed, then

for $x =(4n+2\alpha + 2)\cos^2\varphi$ and $\epsilon\leq \varphi \leq \tfrac{\pi}{2} -\epsilon n^{-1/2}$

:: $e^{-x/2}L^{(\alpha )}_n(x)
=(-1)^{n}(\pi \sin \varphi)^{-1/2}x^{-\alpha/2-1/4}n^{\alpha/2-1/4}
\big\{\sin\left[\left(n+\tfrac{\alpha+1}{2}\right)(\sin 2\varphi-2\varphi)+3\pi/4\right] +(nx)^{-1/2}\mathcal{O}(1)\big\}$

for $x =(4n+2\alpha + 2)\cosh^2\varphi$ and $\epsilon\leq \varphi \leq \omega$

:: $e^{-x/2}L^{(\alpha )}_n(x)
=\tfrac{1}{2}(-1)^{n}(\pi \sinh \varphi )^{-1/2}x^{-\alpha/2-1/4}n^{\alpha /2-1/4}
\exp\left[\left(n+\tfrac{\alpha+1}{2}\right)(2\varphi-\sinh 2\varphi)\right]
\{1+\mathcal{O}\left(n^{-1}\right)\}$

for $x =4n+2\alpha + 2 -2(2n/3)^{1/3}t$ and $t$ complex and bounded

:: $e^{-x/2}L^{(\alpha)}_n(x)
=(-1)^n\pi^{-1}2^{-\alpha-1/3}3^{1/3}n^{-1/3}
\bigg\{A(t)+\mathcal{O}\left(n^{-2/3}\right)\bigg\}$

Literature

{{cite book|first1=Gábor|last1=Szegő|title=Orthogonal polynomials|volume=4|publisher=American Mathematical Society|place=Providence, Rhode Island|date=1975|isbn=0-8218-1023-5}}

References

Category:Analysis

Category:Asymptotic analysis

Category:Orthogonal polynomials