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.[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
denote the n-th Hermite polynomial. Let
and
be positive and fixed, then
- for
and ![{\displaystyle \epsilon \leq \varphi \leq \pi -\epsilon }](https://wikimedia.org/api/rest_v1/media/math/render/svg/be7dc10169486bf10711d7f7769692174d5301b9)
![{\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 \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c5327e7c814aefd0efebaa32294bf2415295a64)
- for
and ![{\displaystyle \epsilon \leq \varphi \leq \omega }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9fcdcb67d96b7c39cf2cfdfec50effe10c22196)
![{\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 \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee96365402da3c116bbee8c83a71ecc69f83bc4d)
- for
and
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 \{}\operatorname {Ai} (t)+{\mathcal {O}}\left(n^{-{2/3}}\right){\bigg \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8924f64adbce9561646a2ef4c531a3d801ce0b92)
where
denotes the Airy function.[3]
(Associated) Laguerre polynomials
Let
denote the n-th associate Laguerre polynomial. Let
be arbitrary and real,
and
be positive and fixed, then
- for
and ![{\displaystyle \epsilon \leq \varphi \leq {\tfrac {\pi }{2}}-\epsilon n^{-1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/93296c518005768d6114f5324d361612e4617a3c)
![{\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 \}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6119fe41d5a8158dcb3367e17825744761b19b35)
- for
and ![{\displaystyle \epsilon \leq \varphi \leq \omega }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9fcdcb67d96b7c39cf2cfdfec50effe10c22196)
![{\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)\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22b07b923bd08a7f942ce6fec0e9faf5456ab485)
- for
and
complex and bounded
.[3]
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.
- ^ Möcklin, Egon (1934). Asymptotische Entwicklungen der Laguerreschen Polynome (Thesis). ETH Zurich. doi:10.3929/ethz-a-000092417.
- ^ a b Szegő, Gábor (1975). Orthogonal polynomials. Vol. 4. Providence, Rhode Island: American Mathematical Society. pp. 200–201. ISBN 0-8218-1023-5.