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Izv. RAN. Ser. Mat., Forthcoming paper (Mi izv9138)  

Plancherel-Rotach type asymptotics for multiple orthogonal Hermite polynomials and recurrence relations

A. I. Aptekareva, S. Yu. Dobrokhotovb, D. N. Tulyakova, A. V. Tsvetkovab*

a Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow
b Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, Moscow

Abstract: We study the asymptotic properties of multiple orthogonal Hermite polynomials, which are determined by the orthogonality relations with respect to two Hermite weights (Gaussian distributions) with shifted maxima. The starting point of the asymptotic analysis is four-term recurrence relations connecting polynomials with adjacent numbers. Asymptotic expansions for consistent growth of the number of the polynomial and its variable (the so-called Plancherel-Rotach type asymptotics) are obtained. Two techniques are used: constructing expansions of basis of homogeneous difference equations and approach based on reducing the difference equation to pseudodifferential equation and using of the theory of Maslov canonical operator. Both approaches give agreed results.

Keywords: Asymptotics, special functions, recurrence relations, pseudodifferential operators, Maslov canonical operator
* Author to whom correspondence should be addressed


English version:
DOI: https://doi.org/10.1070/IM9138

UDC: 517.53
Received: 30.12.2020
Revised: 10.06.2021

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