Uniform Asymptotics of Orthogonal Polynomials Arising from Coherent States

Abstract

In this paper, we study a family of orthogonal polynomials $\{ϕ_n(z)\}$ arising from nonlinear coherent states in quantum optics. Based on the three-term recurrence relation only, we obtain a uniform asymptotic expansion of $ϕ_n(z)$ 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 $ϕ_n(z)$ is provided.