Notes on explicit and inversion formulas for the Chebyshev polynomials of the first two kinds

Feng Qi (1, 2, 3) , Da-Wei Niu (4) , Dongkyu Lim (5)

1 Tianjin Polytechnic University
2 Henan Polytechnic University
3 Inner Mongolia University for Nationalities
4 Henan University of Animal Husbandry and Economy
5 Andong National University

In the paper, starting from the Rodrigues formulas for the Chebyshev polynomials of the first and second kinds, by virtue of the Fa\`a di Bruno formula, with the help of two identities for the Bell polynomials of the second kind, and making use of a new inversion theorem for combinatorial coefficients, the authors derive two nice explicit formulas and their corresponding inversion formulas for the Chebyshev polynomials of the first and second kinds.

Mots clés: explicit formula; inversion formula; Rodrigues formula; Chebyshev polynomial of the first kind; Chebyshev polynomial of the second kind; Faa di Bruno formula; Bell polynomial of the second kind

Domaines: Combinatoire [math.CO]; Théorie des nombres [math.NT]; Analyse classique [math.CA]

Feng Qi, Da-Wei Niu, Dongkyu Lim. Notes on explicit and inversion formulas for the Chebyshev polynomials of the first two kinds: Explicit and inversion formulas for Chebyshev polynomials. Miskolc Mathematical Notes, 2019, 20 (2), pp.1127--1139. ⟨10.18514/MMN.2019.2976⟩. ⟨hal-01705040v2⟩