Geodesic
On generalizations of Chebyshev polynomials and Catalan numbers
Ufa mathematical journal, Tome 13 (2021) no. 2, pp. 8-14 Cet article a éte moissonné depuis la source Math-Net.Ru

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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
Mots-clés : polynomial Pell equation, dessins d'enfants. 
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B. S. Bychkov; G. B. Shabat. On generalizations of Chebyshev polynomials and Catalan numbers. Ufa mathematical journal, Tome 13 (2021) no. 2, pp. 8-14. http://geodesic.mathdoc.fr/item/UFA_2021_13_2_a1/

[1] N. H. Abel, “Sur l'integration de la formule differentielle $\frac{\rho \, \mathrm{d}z}{\sqrt{R}}$, $R $ et $\rho$ etant des fonctions entieres”, J. Reine Angew. Math., 1 (1826), 185–221 | MR | Zbl

[2] J. Harer, D. Zagier, “The Euler characteristic of the moduli space of curves”, Invent. Math., 85 (1986), 457–485 | DOI | MR | Zbl

[3] A. Izosimov, “Pentagrams, inscribed polygons, and Prym varieties”, El. Res. An. in Math. Sc., 23 (2016), 25–40 | MR | Zbl

[4] P. Van Moerbeke, D. Mumford, “The spectrum of difference operators and algebraic curves”, Acta Math., 143 (1979), 93–154 | DOI | MR | Zbl

[5] B. G. Pittel, “Another Proof of the Harer-Zagier Formula”, Electron. J. Comb., 23:1 (2016) | MR

[6] G. Shabat, A. Zvonkin, “Plane trees and algebraic numbers”, Contemporary Math., 178, 1994, 233–275 | DOI | MR | Zbl

[7] A. E. Artisevich, B. S. Bychkov, A. B. Shabat, “Chebyshev polynomials, Catalan numbers, and tridiagonal matrices”, Theor. Math. Phys., 204:1 (2020), 837–842 | DOI | MR | Zbl

[8] N. Vasiliev, A. Zelevinskii, “Chebyshev polynomials and recurrent relations”, Kvant, 1 (1982), 12–19 (in Russian)

[9] V. E. Zakharov, A. B. Shabat, “A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I”, Funct. Anal. Appl., 8:3 (1974), 226–235 | DOI | MR | MR | Zbl

[10] A. Ya. Khinchin, Continued fractions, University of Chicago Press, Chicago, 1964 | MR | Zbl

[11] G. B. Shabat, “Several points of views on Catalan numbers”, Elements of mathematics, eds. A.A. Zaslavskii, A. B. Skopenkov, M. B. Skopenkov, MCCME, M., 2018 (in Russian)