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Ufa Mathematical Journal, 2015, Volume 7, Issue 3, Pages 54–66
DOI: https://doi.org/10.13108/2015-7-3-54
(Mi ufa290)
 

This article is cited in 5 scientific papers (total in 5 papers)

Distribution of zeroes to generalized Hermite polynomials

V. Yu. Novokshenova, A. A. Schelkonogovb

a Institution of Russian Academy of Sciences Institute of Mathematics with Computer Center, Ufa, Russia
b Ufa State Aviation Technical University, Ufa, Russia
References:
Abstract: Asymptotics of the orthogonal polynomial constitute a classic analytic problem. In the paper, we find a distribution of zeroes to generalized Hermite polynomials $H_{m,n}(z)$ as $m=n$, $n\to\infty$, $z=O(\sqrt n)$. These polynomials defined as the Wronskians of classic Hermite polynomials appear in a number of mathematical physics problems as well as in the theory of random matrices. Calculation of asymptotics is based on Riemann–Hilbert problem for Painlevé IV equation which has the solutions $u(z)=-2z +\partial_z\ln H_{m,n+1}(z)/H_{m+1,n}(z)$. In this scaling limit the Riemann-Hilbert problem is solved in elementary functions. As a result, we come to analogs of Plancherel–Rotach formulas for asymptotics of classical Hermite polynomials.
Keywords: generalized Hermite polynomials, Painlevé IV equation, meromorphic solutions, distribution of zeroes, Riemann–Hilbert problem, Deift–Zhou method, Plancherel–Rotach formulas.
Received: 24.08.2015
Bibliographic databases:
Document Type: Article
UDC: 517.587+517.923
Language: English
Original paper language: Russian
Citation: V. Yu. Novokshenov, A. A. Schelkonogov, “Distribution of zeroes to generalized Hermite polynomials”, Ufa Math. J., 7:3 (2015), 54–66
Citation in format AMSBIB
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\by V.~Yu.~Novokshenov, A.~A.~Schelkonogov
\paper Distribution of zeroes to generalized Hermite polynomials
\jour Ufa Math. J.
\yr 2015
\vol 7
\issue 3
\pages 54--66
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\crossref{https://doi.org/10.13108/2015-7-3-54}
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  • https://doi.org/10.13108/2015-7-3-54
  • https://www.mathnet.ru/eng/ufa/v7/i3/p57
  • This publication is cited in the following 5 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
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