Geodesic
Distribution of zeroes to generalized Hermite polynomials
Ufa mathematical journal, Tome 7 (2015) no. 3, pp. 54-66

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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, distribution of zeroes, Riemann–Hilbert problem, Deift–Zhou method
Mots-clés : meromorphic solutions, Plancherel–Rotach formulas. 
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     author = {V. Yu. Novokshenov and A. A. Schelkonogov},
     title = {Distribution of zeroes to generalized {Hermite} polynomials},
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     pages = {54--66},
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     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2015_7_3_a6/}
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V. Yu. Novokshenov; A. A. Schelkonogov. Distribution of zeroes to generalized Hermite polynomials. Ufa mathematical journal, Tome 7 (2015) no. 3, pp. 54-66. http://geodesic.mathdoc.fr/item/UFA_2015_7_3_a6/