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.^[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 ${\displaystyle H_{n}(x)}$ denote the n-th Hermite polynomial. Let ${\displaystyle \epsilon }$ and ${\displaystyle \omega }$ be positive and fixed, then

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

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

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

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

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

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

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

(Associated) Laguerre polynomials

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

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

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

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

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

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

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

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

Literature

• Szegő, Gábor (1975). Orthogonal polynomials. Vol. 4. Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-1023-5.