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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.
† Author to whom correspondence should be addressed
DOI:
https://doi.org/10.4213/sm8416
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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
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http://mi.mathnet.ru/eng/msb8416https://doi.org/10.4213/sm8416 http://mi.mathnet.ru/eng/msb/v205/i12/p17
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A. I. Aptekarev, D. N. Tulyakov, “The Saturation Regime of Meixner Polynomials and the Discrete Bessel Kernel”, Math. Notes, 98:1 (2015), 180–184
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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
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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
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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
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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
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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.
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