Geodesic
On generalizations of Chebyshev polynomials and Catalan numbers
Ufa mathematical journal, Tome 13 (2021) no. 2, pp. 8-14

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{UFA_2021_13_2_a1,
     author = {B. S. Bychkov and G. B. Shabat},
     title = {On generalizations of {Chebyshev} polynomials and {Catalan} numbers},
     journal = {Ufa mathematical journal},
     pages = {8--14},
     publisher = {mathdoc},
     volume = {13},
     number = {2},
     year = {2021},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2021_13_2_a1/}
}
TY  - JOUR
AU  - B. S. Bychkov
AU  - G. B. Shabat
TI  - On generalizations of Chebyshev polynomials and Catalan numbers
JO  - Ufa mathematical journal
PY  - 2021
SP  - 8
EP  - 14
VL  - 13
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/UFA_2021_13_2_a1/
LA  - en
ID  - UFA_2021_13_2_a1
ER  - 
%0 Journal Article
%A B. S. Bychkov
%A G. B. Shabat
%T On generalizations of Chebyshev polynomials and Catalan numbers
%J Ufa mathematical journal
%D 2021
%P 8-14
%V 13
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/UFA_2021_13_2_a1/
%G en
%F UFA_2021_13_2_a1
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/