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Mat. Sb., 2009, Volume 200, Number 5, Pages 129–158 (Mi msb6393)  

This article is cited in 12 scientific papers (total in 12 papers)

Difference equations having bases with powerlike growth which are perturbed by a spectral parameter

D. N. Tulyakov

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

Abstract: The asymptotic behaviour of solutions with powerlike growth of recurrence relations with a spectral parameter is investigated. A class of recurrence relations in which all basis solutions have powerlike growth is introduced. Recurrence relations in this class are linearly perturbed by a spectral parameter; for solutions of the new recurrence relations asymptotic formulae are obtained which are uniform with respect to the spectral parameter ranging within appropriate bounds. The theorems obtained are used for deriving new local asymptotic formulae for orthogonal and multiple orthogonal polynomials in a neighbourhood of the end-points of the support of the orthogonality weights.
Bibliography: 14 titles.

Keywords: asymptotic behaviour of solutions of recurrence relations, local asymptotics of orthogonal and multiple orthogonal polynomials, Poincaré's theorem, Perron's theorem, Birkhoff-Trjitzinsky theorem.


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English version:
Sbornik: Mathematics, 2009, 200:5, 753–781

Bibliographic databases:

UDC: 517.929+517.538.3+517.538.6
MSC: 39A11, 42C05
Received: 14.07.2008 and 24.02.2009

Citation: D. N. Tulyakov, “Difference equations having bases with powerlike growth which are perturbed by a spectral parameter”, Mat. Sb., 200:5 (2009), 129–158; Sb. Math., 200:5 (2009), 753–781

Citation in format AMSBIB
\by D.~N.~Tulyakov
\paper Difference equations having bases with powerlike growth which are perturbed by a~spectral parameter
\jour Mat. Sb.
\yr 2009
\vol 200
\issue 5
\pages 129--158
\jour Sb. Math.
\yr 2009
\vol 200
\issue 5
\pages 753--781

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    This publication is cited in the following articles:
    1. D. N. Tulyakov, “Plancherel-Rotach type asymptotics for solutions of linear recurrence relations with rational coefficients”, Sb. Math., 201:9 (2010), 1355–1402  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. A. I. Aptekarev, V. I. Buslaev, A. Martínez-Finkelshtein, S. P. Suetin, “Padé approximants, continued fractions, and orthogonal polynomials”, Russian Math. Surveys, 66:6 (2011), 1049–1131  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. K. Deschout, A. B. J. Kuijlaars, “Critical behavior in Angelesco ensembles”, J. Math. Phys, 53:12 (2012), 123523  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    4. A. I. Aptekarev, D. N. Tulyakov, “Glavnyi chlen asimptotiki reshenii chetyrekhchlennykh rekursii”, Preprinty IPM im. M. V. Keldysha, 2013, 001, 20 pp.  mathnet
    5. Dominici D., “Mehler–Heine type formulas for Charlier and Meixner polynomials”, Ramanujan J., 39:2 (2016), 271–289  crossref  mathscinet  zmath  isi  scopus
    6. Draux A., “-Coherent pairs of linear functionals and Markov-Bernstein inequalities”, J. Differ. Equ. Appl., 22:11 (2016), 1583–1608  crossref  mathscinet  zmath  isi  scopus
    7. Van Assche W., “Mehler-Heine asymptotics for multiple orthogonal polynomials”, Proc. Amer. Math. Soc., 145:1 (2017), 303–314  crossref  mathscinet  zmath  isi  scopus
    8. Aptekarev A.I. Van Assche W. Yattselev M.L., “Hermite-Padé Approximants for a Pair of Cauchy Transforms with Overlapping Symmetric Supports”, Commun. Pure Appl. Math., 70:3 (2017), 444–510  crossref  mathscinet  zmath  isi  scopus
    9. A. I. Aptekarev, D. N. Tulyakov, “Koordinaty Khesse dlya odnoi algebraicheskoi krivoi tretego poryadka”, Preprinty IPM im. M. V. Keldysha, 2017, 054, 16 pp.  mathnet  crossref
    10. A. I. Aptekarev, A. Draux, D. N. Tulyakov, “On asymptotics of the sharp constants of the Markov–Bernshtein inequalities for the Sobolev spaces with coherent weights”, Preprinty IPM im. M. V. Keldysha, 2017, 059, 20 pp.  mathnet  crossref
    11. Aptekarev A.I. Draux A. Tulyakov D.N., “On Asymptotics of the Sharp Constants of the Markov-Bernshtein Inequalities For the Sobolev Spaces”, Lobachevskii J. Math., 39:5 (2018), 609–622  mathnet  crossref  mathscinet  zmath  isi  scopus
    12. 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  elib
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