The Plancherel-Rotach formula for Chebyshev-Hermite functions on half-intervals contracting to infinity. (English. Russian original) Zbl 1034.42023
Math. Notes 72, No. 1, 66-74 (2002); translation from Mat. Zametki 72, No. 1, 74-83 (2002).
The author proves the asymptotic formula
\[
(-1)^ne^{s^2/2}\frac{(e^{-s^2})^{(n)}}{\sqrt{2^nn!\sqrt{\pi}}}=\frac{1}{(2\pi)^{1/2}(s^2 -(2n+1)^{1/4}}
\]
\[ \times\exp\left(-\frac{s}{2}\sqrt{s^2-(2n+1)}+\frac{2n+1}{2}\log\left(\frac{s+\sqrt{s^2-(2n+1)}}{\sqrt{2n+1}}\right)\right) \]
\[ \times\frac{1+\alpha(s,2n+1)}{(1+\alpha(+\infty,2n+1))(1+\varepsilon(n))} \] where \(s\in(\sqrt{\lambda+(50\lambda/9)^{1/3}},+\infty)\) and there are given esimates for \(\alpha(s,2n+1)\) and \(\varepsilon(n)\) \((\alpha(+\infty,2n+1)=\lim_{s\to +\infty}\alpha(s,2n+1)).\)
\[ \times\exp\left(-\frac{s}{2}\sqrt{s^2-(2n+1)}+\frac{2n+1}{2}\log\left(\frac{s+\sqrt{s^2-(2n+1)}}{\sqrt{2n+1}}\right)\right) \]
\[ \times\frac{1+\alpha(s,2n+1)}{(1+\alpha(+\infty,2n+1))(1+\varepsilon(n))} \] where \(s\in(\sqrt{\lambda+(50\lambda/9)^{1/3}},+\infty)\) and there are given esimates for \(\alpha(s,2n+1)\) and \(\varepsilon(n)\) \((\alpha(+\infty,2n+1)=\lim_{s\to +\infty}\alpha(s,2n+1)).\)
Reviewer: Alexei Lukashov (Saratov)
MSC:
| 42C05 | Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
