Plancherel–Rotach asymptotics

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

• for $x=(2n+1{)}^{1/2}\cos \phi$ and $ϵ\le \phi \le \pi -ϵ$

$e^{-x^2/2}{H}_n(x)=2^{n/2+1/4}(n!{)}^{1/2}(\pi n{)}^{-1/4}(\sin \phi {)}^{-1/2}\{\sin [(\frac{n}{2}+\frac{1}{4})(\sin 2\phi -2\phi )+3\frac{\pi }{4}]+𝒪(n^{-1})\}$

• for $x=(2n+1{)}^{1/2}\cosh \phi$ and $ϵ\le \phi \le \omega$

$e^{-x^2/2}{H}_n(x)=2^{n/2-3/4}(n!{)}^{1/2}(\pi n{)}^{-1/4}(\sinh \phi {)}^{-1/2}\exp [(\frac{n}{2}+\frac{1}{4})(2\phi -\sinh 2\phi )]\{1+𝒪(n^{-1})\}$

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

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

where $A(t)=3^{-1/3}\pi \mathrm{Ai}(-3^{-1/3}t)$ and $\mathrm{Ai}$ denotes the Airy function.

Probabalist's Hermite polynomials

Let ${\mathrm{He}}_n(x)$ denote the n-th probabalist's Hermite polynomial. These are related to the (physicist's) Hermite polynomials $H_n(x)$ by the relations $H_n(x)=2^{\frac{n}{2}}{\mathrm{He}}_n\left(\sqrt{2}x\right),{\mathrm{He}}_n(x)=2^{-\frac{n}{2}}{H}_n\left(\frac{x}{\sqrt{2}}\right).$

Three approximations for the probabilists Hermite polynomials illustrated when n=25

The Plancherel–Rotach asymptotics for ${\mathrm{He}}_n(x)$ can be written as:

• [Small $x$ regime] For fixed $\alpha \in (0,\pi /2)$, set $x=2\cos (\alpha )\sqrt{n}$ and then as $n\to \mathrm{\infty }$ one has

$$\begin{array}{cc}{\mathrm{He}}_n\left(x\right)& =\sqrt{\frac{2n^n}{\sin (\alpha )}}{e}^{n({\cos }^2\alpha -\frac{1}{2})}\cos (n(\sin \alpha \cos \alpha -\alpha )-\frac{1}{2}(\alpha +\frac{\pi }{2}))(1+o(1))\end{array}$$

• [Large x regime] For fixed $\gamma >0$, set $x=2\cosh (\gamma )\sqrt{n}$. Then as $n\to \mathrm{\infty }$ one has $$\begin{array}{cc}{\mathrm{He}}_n\left(x\right)& =n^{n/2}{e}^{n(\gamma +\frac{1}{2}{e}^{-2\gamma })}\frac{e^{\gamma /2}}{\sqrt{2\sinh (\gamma )}}(1+o(1)).\end{array}$$

• [Edge regime] For fixed $u\in ℝ$, set $x=2\sqrt{n}+n^{-1/6}u$. Then as $n\to \mathrm{\infty }$ one has

$$\begin{array}{cc}{\mathrm{He}}_n\left(x\right)& =\sqrt{2\pi }{e}^{n/2}{n}^{n/2}{n}^{1/6}{e}^{u{n}^{1/3}}\mathrm{Ai}(u)(1+o(1)).\end{array}$$

(Associated) Laguerre polynomials

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

• for $x=(4n+2\alpha +2){\cos }^2\phi$ and $ϵ\le \phi \le \frac{\pi }{2}-ϵn^{-1/2}$

$e^{-x/2}{L}_n^{(\alpha )}(x)=(-1{)}^n(\pi \sin \phi {)}^{-1/2}{x}^{-\alpha /2-1/4}{n}^{\alpha /2-1/4}\{\sin [(n+\frac{\alpha +1}{2})(\sin 2\phi -2\phi )+3\pi /4]+(nx{)}^{-1/2}𝒪(1)\}$

• for $x=(4n+2\alpha +2){\cosh }^2\phi$ and $ϵ\le \phi \le \omega$

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

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

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

where $A(t)=\pi \mathrm{Ai}(-3^{-1/3}t)$ and $\mathrm{Ai}$ denotes the Airy function.

Literature

• Template:Cite book
• Template:Cite journal