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Ufimsk. Mat. Zh., 2015, Volume 7, Issue 3, Pages 57–69 (Mi ufa290)  

This article is cited in 3 scientific papers (total in 3 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

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.

Full text: PDF file (319 kB)
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English version:
Ufa Mathematical Journal, 2015, 7:3, 54–66 (PDF, 415 kB); https://doi.org/10.13108/2015-7-3-54

Bibliographic databases:

UDC: 517.587+517.923
MSC: 30D35, 30E10, 33C75, 34M35, 34M55, 34M60
Received: 24.08.2015

Citation: V. Yu. Novokshenov, A. A. Schelkonogov, “Distribution of zeroes to generalized Hermite polynomials”, Ufimsk. Mat. Zh., 7:3 (2015), 57–69; 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 Ufimsk. Mat. Zh.
\yr 2015
\vol 7
\issue 3
\pages 57--69
\mathnet{http://mi.mathnet.ru/ufa290}
\elib{http://elibrary.ru/item.asp?id=24716954}
\transl
\jour Ufa Math. J.
\yr 2015
\vol 7
\issue 3
\pages 54--66
\crossref{https://doi.org/10.13108/2015-7-3-54}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84959046973}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. V. Yu. Novokshenov, “Discrete integrable equations and special functions”, Ufa Math. J., 9:3 (2017), 118–130  mathnet  crossref  isi  elib
    2. P. Roffelsen, D. Masoero, “Poles of Painlevé IV Rationals and their Distribution”, SIGMA, 14 (2018), 002, 49 pp.  mathnet  crossref
    3. V. Yu. Novokshenov, “Generalized Hermite Polynomials and Monodromy-Free Schrödinger Operators”, SIGMA, 14 (2018), 106, 13 pp.  mathnet  crossref
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