Uniform asymptotics for the discrete Laguerre polynomials
| Autor: | Dai, Dan, Yao, Luming |
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| Rok vydání: | 2021 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | In this paper, we consider the discrete Laguerre polynomials $P_{n, N}(z)$ orthogonal with respect to the weight function $w(x) = x^{\alpha} e^{-N cx}$ supported on the infinite nodes $L_N = \{ x_{k,N} = \frac{k^2}{N^2}, k \in \mathbb{N} \}$. We focus on the "band-saturated region" situation when the parameter $c > \frac{\pi^2}{4}$. As $n \to \infty$, uniform expansions for $P_{n, n}(z)$ are achieved for $z$ in different regions in the complex plane. Typically, the Airy-function expansions and Gamma-function expansions are derived for $z$ near the endpoints of the band and the origin, respectively. The asymptotics for the normalizing coefficient $h_{n, N}$, recurrence coefficients $\mathscr{B}_{n, N}$ and $\mathscr{A}_{n, N}^2$, are also obtained. Our method is based on the Deift-Zhou steepest descent method for Riemann-Hilbert problems. Comment: 37 pages, 6 figures |
| Databáze: | arXiv |
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