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Mat. Sb., 2010, Volume 201, Number 9, Pages 111–158 (Mi msb7618)  

This article is cited in 16 scientific papers (total in 16 papers)

Plancherel-Rotach type asymptotics for solutions of linear recurrence relations with rational coefficients

D. N. Tulyakov

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

Abstract: The asymptotic behaviour of solutions of difference equations with respect to the variable $n$ with spectral parameter $x$ is investigated. A new method for finding asymptotic expansions for basis solutions in overlapping domains of the $(n,x)$-space which extend to infinity is proposed. In principle, matching the expansions in the intersection of these domains makes it possible to determine the global asymptotic picture of the behaviour of solutions of equations in the complex plane of the spectral parameter $x$ for suitable scaling depending on $n$. The potential of the method is demonstrated using the examples of the Hermite and Meixner polynomials.
Bibliography: 27 titles.

Keywords: recurrence relations, asymptotic behaviour of solutions of difference equations, orthogonal polynomials, Hermite polynomials, Meixner polynomials.

DOI: https://doi.org/10.4213/sm7618

Full text: PDF file (860 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2010, 201:9, 1355–1402

Bibliographic databases:

UDC: 517.929+517.53
MSC: Primary 11B37, 30E15; Secondary 42A52
Received: 05.08.2009 and 15.06.2010

Citation: D. N. Tulyakov, “Plancherel-Rotach type asymptotics for solutions of linear recurrence relations with rational coefficients”, Mat. Sb., 201:9 (2010), 111–158; Sb. Math., 201:9 (2010), 1355–1402

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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, “Asymptotic regimes in saturation zones for the C-D-kernels for an ensemble of Meixner orthogonal polynomials”, Russian Math. Surveys, 66:1 (2011), 173–175  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. A. I. Aptekarev, D. N. Tulyakov, “Asymptotics of Meixner polynomials and Christoffel-Darboux kernels”, Trans. Moscow Math. Soc., 73 (2012), 67–106  mathnet  crossref  mathscinet  zmath  elib
    3. A. V. Komlov, S. P. Suetin, “An asymptotic formula for polynomials orthonormal with respect to a varying weight”, Trans. Moscow Math. Soc., 73 (2012), 139–159  mathnet  crossref  mathscinet  zmath  elib
    4. A. I. Aptekarev, D. N. Tulyakov, “Asymptotics of $L_p$-norms of Hermite polynomials”, Preprinty IPM im. M. V. Keldysha, 2012, 070, 22 pp.  mathnet
    5. M. E. H. Ismail, Xin Li, “Plancherel–Rotach asymptotics for $q$-orthogonal polynomials”, Constr. Approx., 37:3 (2013), 341–356  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    6. A. I. Aptekarev, D. N. Tulyakov, “Glavnyi chlen asimptotiki reshenii chetyrekhchlennykh rekursii”, Preprinty IPM im. M. V. Keldysha, 2013, 001, 20 pp.  mathnet
    7. A. Aptekarev, J. Arvesú, “Asymptotics for multiple Meixner polynomials”, J. Math. Anal. Appl., 411:2 (2014), 485–505  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    8. B. X. Fejzullahu, “Mehler-Heine formulas for orthogonal polynomials with respect to the modified Jacobi weight”, Proc. Amer. Math. Soc., 142:6 (2014), 2035–2045  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    9. J. S. Dehesa, A. Guerrero, J. L. López, P. Sánchez-Moreno, “Asymptotics $(p\to\infty)$ of $L_p$-norms of hypergeometric orthogonal polynom”, J. Math. Chem., 52:1 (2014), 283–300  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    10. A. I. Aptekarev, D. N. Tulyakov, “The leading term of the Plancherel-Rotach asymptotic formula for solutions of recurrence relations”, Sb. Math., 205:12 (2014), 1696–1719  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    11. 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
    12. A. I. Aptekarev, D. N. Tulyakov, “Asymptotics of $L_p$-norms of Laguerre polynomials and entropic moments of $D$-dimensional oscillator”, Preprinty IPM im. M. V. Keldysha, 2015, 041, 15 pp.  mathnet
    13. A. I. Aptekarev, D. N. Tulyakov, “Asymptotics of $L_p$-norms of Laguerre polynomials”, Russian Math. Surveys, 70:5 (2015), 955–957  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    14. 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
    15. 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
    16. S. Yu. Dobrokhotov, A. V. Tsvetkova, “Lagrangian Manifolds Related to the Asymptotics of Hermite Polynomials”, Math. Notes, 104:6 (2018), 810–822  mathnet  crossref  crossref  isi  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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