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Teoreticheskaya i Matematicheskaya Fizika, 2020, Volume 204, Number 1, Pages 3–9
DOI: https://doi.org/10.4213/tmf9885 (Mi tmf9885) 		 
This article is cited in 3 scientific papers (total in 3 papers)

Chebyshev polynomials, Catalan numbers, and tridiagonal matrices

A. E. Artisevicha, B. S. Bychkovb, A. B. Shabatc

a Adyghe State University, Maykop, Russia
b National Research University "Higher School of Economics", Moscow, Russia
c Landau Institute for Theoretical Physics of Russian Academy of Sciences, Moscow, Russia
Full-text PDF (399 kB) Citations (3)
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DOI: https://doi.org/10.4213/tmf9885
Abstract: We establish a relation between linear second-order difference equations corresponding to Chebyshev polynomials and Catalan numbers. The latter are the limit coefficients of a converging series of rational functions corresponding to the Riccati equation. As the main application, we show a relation between the polynomials φn(μ) that are solutions of the problem of commutation of a tridiagonal matrix with the simplest Vandermonde matrix and Chebyshev polynomials.
Keywords: Chebyshev polynomial, tridiagonal matrix.
Funding agency Grant number
Ministry of Science and Higher Education of the Russian Federation
The research of B. S. Bychkov was performed within a project supported by the Ministry of Science and Higher Education of the Russian Federation (CIS at Demidov Yaroslavl State University).
Received: 27.01.2020
Revised: 27.01.2020
English version:
Theoretical and Mathematical Physics, 2020, Volume 204, Issue 1, Pages 837–842
DOI: https://doi.org/10.1134/S0040577920070016
Bibliographic databases:
Document Type: Article
Language: Russian
Citation: A. E. Artisevich, B. S. Bychkov, A. B. Shabat, “Chebyshev polynomials, Catalan numbers, and tridiagonal matrices”, TMF, 204:1 (2020), 3–9; Theoret. and Math. Phys., 204:1 (2020), 837–842
Citation in format AMSBIB
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  • https://www.mathnet.ru/eng/tmf9885
  • https://doi.org/10.4213/tmf9885
  • https://www.mathnet.ru/eng/tmf/v204/i1/p3
    • This publication is cited in the following 3 articles:
    1. Mariusz Białecki, “Simple Rules of a Discrete Stochastic Process Leading to Catalan-like Recurrences”, Algorithms, 18:3 (2025), 149  crossref
    2. V.E. Adler, “Bogoyavlensky Lattices and Generalized Catalan Numbers”, Russ. J. Math. Phys., 31:1 (2024), 1  crossref  mathscinet
    3. B. S. Bychkov, G. B. Shabat, “On generalizations of Chebyshev polynomials and Catalan numbers”, Ufa Math. J., 13:2 (2021), 8–14  mathnet  crossref  isi
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    		Теоретическая и математическая физика Theoretical and Mathematical Physics
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