Uniform Asymptotics of Orthogonal Polynomials Arising from Coherent States

TL;DR

This paper derives uniform asymptotic expansions for a family of orthogonal polynomials from quantum optics, revealing unique singularities and zero distributions, using the Wang and Wong difference equation method.

Contribution

It provides the first uniform asymptotic analysis of these polynomials based solely on their recurrence relation, uncovering novel singularity behavior.

Findings

Asymptotic expansion of polynomials as degree n grows

Identification of unusual singularity in the weight function

Determination of the zero distribution of the polynomials

Abstract

In this paper, we study a family of orthogonal polynomials $\{\phi_n(z)\}$ arising from nonlinear coherent states in quantum optics. Based on the three-term recurrence relation only, we obtain a uniform asymptotic expansion of $\phi_n(z)$ as the polynomial degree $n$ tends to infinity. Our asymptotic results suggest that the weight function associated with the polynomials has an unusual singularity, which has never appeared for orthogonal polynomials in the Askey scheme. Our main technique is the Wang and Wong's difference equation method. In addition, the limiting zero distribution of the polynomials $\phi_n(z)$ is provided.