On the constant coefficients of a certain recurrence relation: A simple proof

Milica Anđelić; Carlos M da Fonseca

doi:10.1016/j.heliyon.2021.e07764

On the constant coefficients of a certain recurrence relation: A simple proof

Heliyon. 2021 Aug 13;7(8):e07764. doi: 10.1016/j.heliyon.2021.e07764. eCollection 2021 Aug.

Authors

Milica Anđelić¹, Carlos M da Fonseca^{2

3}

Affiliations

¹ Department of Mathematics, Kuwait University, Safat 13060, Kuwait.
² Kuwait College of Science and Technology, Doha District, Safat 13133, Kuwait.
³ Chair of Computational Mathematics, University of Deusto, 48007 Bilbao, Basque Country, Spain.

Abstract

In this note, we provide a short proof for the explicit formulas of the coefficients of a particular 3-term recurrence relation derived from a k-periodic recurrence. Any of the recurrences can be naturally interpreted in terms of determinants of Hessenberg matrices families.

Keywords: Chebyshev polynomials of second kind; Determinant; Recurrence relation; Tridiagonal k-Toeplitz matrices.