Weights with both absolutely continuous and discrete components: Asymptotics via the Riemann-Hilbert approach

2014/12/30 by Xiao‐Bo Wu, Xiao-Bo Wu, Wu, Xiao-Bo +8
Computer Science · Mathematics · Physics and Astronomy · #33C10 #33C45 #41A60 #Advanced Mathematical Theories and Applications #Complex Variables (math.CV) #FOS: Mathematics #Mathematical functions and polynomials #Matrix Theory and Algorithms #math.CV #msc:33C10 #msc:33C45 #msc:41A60

paper · pdf · doi:10.48550/arxiv.1412.8580

31 pages, 5 figures

arxiv created 2014/12/30 · openalex publication_date 2014/12/30 · arxiv updated 2014/12/31 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

We study the uniform asymptotics for the orthogonal polynomials with respect to weights composed of both absolutely continuous measure and discrete measure, by taking a special class of the sieved Pollazek Polynomials as an example. The Plancherel-Rotach type asymptotics of the sieved Pollazek Polynomials are obtained in the whole complex plane. The Riemann-Hilbert method is applied to derive the results. A main feature of the treatment is the appearance of a new band consisting of two adjacent intervals, one of which is a portion of the support of the absolutely continuous measure, the other is the discrete band.

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