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Ufa Mathematical Journal, 2021, Volume 13, Issue 2, Pages 8–14
DOI: https://doi.org/10.13108/2021-13-2-8
(Mi ufa558)
 

This article is cited in 1 scientific paper (total in 1 paper)

On generalizations of Chebyshev polynomials and Catalan numbers

B. S. Bychkovab, G. B. Shabatc

a P.G. Demidov Yaroslavl State University, Sovetskaya str. 14, 150003, Yaroslavl, Russia
b HSE University, Usacheva str. 6, 119048, Moscow, Russia
c Russian State University for The Humanities, Miusskaya sq. 6, 125993, Moscow, Russia
References:
Abstract: We provide possible directions of generalizations of earlier found relations between the Chebyshev polynomials and the Catalan numbers arising in studying commuting difference operators. These generalizations are mostly related with ideas proposed by N.H. Abel in his publication in 1826, which then were reproduced by many authors in a modern language. As generalization of Chebyshev polynomials, we propose to consider polynomials with exactly two critical values well-studied in a so-called theory of dessins d'enfants. The Catalan numbers are located in the first column of the table of Harer–Zagier numbers related with the distribution by genus of orientable sewing of polygons with even number of sides. The commuting difference operators are implicitly contained in the Abel theory, who studied quasi-elliptic integrals, namely, the elliptic integrals of 3rd kind integrable in terms of logarithms. In the present work we formulate conjectures on relation between the main Abel theorem and commuting semi-infinite matrices. In the work we provide calculations supporting the conjectured relations.
Keywords: Chebyshev polynomials, Catalan numbers, Harer-Zagier numbers, polynomial Pell equation, dessins d'enfants.
Funding agency Grant number
Foundation for the Development of Theoretical Physics and Mathematics BASIS
Simons Foundation
The work is supported by the grant of Theoretical Physics and Mathematics Advancement Foundation “BASIS”. The research by G.B. Shabat is supported by Simons foundation.
Received: 21.04.2021
Russian version:
Ufimskii Matematicheskii Zhurnal, 2021, Volume 13, Issue 2, Pages 11–17
Bibliographic databases:
Document Type: Article
UDC: 517.587, 517.929
MSC: 39A70, 33C75
Language: English
Original paper language: Russian
Citation: B. S. Bychkov, G. B. Shabat, “On generalizations of Chebyshev polynomials and Catalan numbers”, Ufimsk. Mat. Zh., 13:2 (2021), 11–17; Ufa Math. J., 13:2 (2021), 8–14
Citation in format AMSBIB
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\by B.~S.~Bychkov, G.~B.~Shabat
\paper On generalizations of Chebyshev polynomials and Catalan numbers
\jour Ufimsk. Mat. Zh.
\yr 2021
\vol 13
\issue 2
\pages 11--17
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\transl
\jour Ufa Math. J.
\yr 2021
\vol 13
\issue 2
\pages 8--14
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  • https://doi.org/10.13108/2021-13-2-8
  • https://www.mathnet.ru/eng/ufa/v13/i2/p11
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    References:33
     
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