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Mat. Sb., 2014, Volume 205, Number 12, Pages 17–40 (Mi msb8416)  

This article is cited in 5 scientific papers (total in 6 papers)

The leading term of the Plancherel-Rotach asymptotic formula for solutions of recurrence relations

A. I. Aptekarev, D. N. Tulyakov

M. V. Keldysh Institute for Applied Mathematics, Russian Academy of Sciences, Moscow

Abstract: Recurrence relations generating Padé and Hermite-Padé polynomials are considered. Their coefficients increase with the index of the relation, but after dividing by an appropriate power of the scaling function they tend to a finite limit. As a result, after scaling the polynomials ‘stabilize’ for large indices; this type of asymptotic behaviour is called Plancherel-Rotach asymptotics. An explicit expression for the leading term of the asymptotic formula, which is valid outside sets containing the zeros of the polynomials, is obtained for wide classes of three- and four-term relations. For three-term recurrence relations this result generalizes a theorem Van Assche obtained for recurrence relations with ‘regularly’ growing coefficients.
Bibliography: 19 titles.

Keywords: high-order recurrence relations, multiple orthogonal polynomials, Hermite-Padé approximants, difference operators.

Funding Agency Grant Number
Russian Science Foundation 14-21-00025

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/sm8416

Full text: PDF file (616 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2014, 205:12, 1696–1719

Bibliographic databases:

UDC: 517.53
MSC: 39A06
Received: 25.08.2014 and 21.10.2014

Citation: A. I. Aptekarev, D. N. Tulyakov, “The leading term of the Plancherel-Rotach asymptotic formula for solutions of recurrence relations”, Mat. Sb., 205:12 (2014), 17–40; Sb. Math., 205:12 (2014), 1696–1719

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. I. Aptekarev, D. N. Tulyakov, “The Saturation Regime of Meixner Polynomials and the Discrete Bessel Kernel”, Math. Notes, 98:1 (2015), 180–184  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    2. V. I. Buslaev, S. P. Suetin, “On equilibrium problems related to the distribution of zeros of the Hermite–Padé polynomials”, Proc. Steklov Inst. Math., 290:1 (2015), 256–263  mathnet  crossref  crossref  isi  elib  elib
    3. V. M. Buchstaber, V. N. Dubinin, V. A. Kaliaguine, B. S. Kashin, V. N. Sorokin, S. P. Suetin, D. N. Tulyakov, B. N. Chetverushkin, E. M. Chirka, A. A. Shkalikov, “Alexander Ivanovich Aptekarev (on his 60th birthday)”, Russian Math. Surveys, 70:5 (2015), 965–973  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. A. I. Aptekarev, “The Mhaskar-Saff variational principle and location of the shocks of certain hyperbolic equations”, Moder trends in constructive function theory, Conference and School on Constructive Functions in honor of Ed Saff's 70th Birthday (Vanderbilt Univ, Nashville, 2014), Contemp. Math., 661, Amer. Math. Soc., Providence, RI, 2016, 167–186  crossref  mathscinet  zmath  isi
    5. A. I. Aptekarev, M. A. Lapik, Yu. N. Orlov, “Asymptotic behavior of the spectrum of combination scattering at Stokes phonons”, Theoret. and Math. Phys., 193:1 (2017), 1480–1497  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    6. A. I. Aptekarev, D. N. Tulyakov, “Asimptoticheskii bazis reshenii $q$-rekurrentnykh sootnoshenii vne zony blizkikh sobstvennykh znachenii”, Preprinty IPM im. M. V. Keldysha, 2018, 159, 24 pp.  mathnet  crossref
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