### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 070, 17 pages      arXiv:1408.5654      https://doi.org/10.3842/SIGMA.2015.070

### Uniform Asymptotics of Orthogonal Polynomials Arising from Coherent States

Dan Dai a, Weiying Hu a and Xiang-Sheng Wang b
a) Department of Mathematics, City University of Hong Kong, Hong Kong
b) Department of Mathematics, Southeast Missouri State University, Cape Girardeau, MO 63701, USA

Received April 01, 2015, in final form August 25, 2015; Published online August 31, 2015

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.

Key words: uniform asymptotics; orthogonal polynomials; coherent states; three-term recurrence relation.

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