Affiliation:
1. School of Mathematical Sciences, Huaqiao University 1 , Quanzhou 362021, China
2. Department of Mathematics, Faculty of Science and Technology, University of Macau 2 , Macau, China
Abstract
We study the monic polynomials Pn(x; t), orthogonal with respect to a symmetric perturbed Gaussian weight function w(x)=w(x;t)≔e−x21+tx2λ,x∈R, with t>0,λ∈R. This problem is related to single-user multiple-input multiple-output systems in information theory. It is shown that the recurrence coefficient βn(t) is related to a particular Painlevé V transcendent, and the sub-leading coefficient p(n, t) of Pn(x; t) (Pn(x; t) = xn + p(n, t)xn−2 + ⋯) satisfies the Jimbo–Miwa–Okamoto σ-form of the Painlevé V equation. Furthermore, we derive the second-order difference equations satisfied by βn(t) and p(n, t), respectively. This enables us to obtain the large n full asymptotic expansions for βn(t) and p(n, t) with the aid of Dyson’s Coulomb fluid approach in the one-cut case [i.e., λt ≤ 1 (t > 0)]. We also consider the Hankel determinant Dn(t), generated by the perturbed Gaussian weight. It is found that Φn(t), a quantity allied to the logarithmic derivative of Dn(t) via Φn(t)=2t2ddtlnDn(t)−2nλt, can be expressed in terms of βn(t) and p(n, t). Based on this result, we obtain the large n asymptotic expansion of Φn(t) and then that of the Hankel determinant Dn(t) in the one-cut case.
Funder
National Natural Science Foundation of China
Fundamental Research Funds for the Central Universities
Scientific Research Funds of Huaqiao University
Macau Science and Technology Development Fund
Subject
Mathematical Physics,Statistical and Nonlinear Physics
Reference43 articles.
1. Encyclopedia of Mathematics and its Applications Vol. 98,2005
2. On characterizations of classical polynomials;J. Comput. Appl. Math.,2006
3. Perturbed Laguerre unitary ensembles, Hankel determinants, and information theory;Math. Methods Appl. Sci.,2015
Cited by
1 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献