Determinantal Formulas and Recurrent Relations for Bi-Periodic Fibonacci and Lucas Polynomials

Abstract

In the paper, the authors find closed formulas and recurrent relations for bi-periodic Fibonacci polynomials and for bi-periodic Lucas polynomials in terms of the Hessenberg determinants. Consequently, the authors derive closed formulas and recurrent relations for the Fibonacci, Lucas, bi-periodic Fibonacci, and bi-periodic Lucas numbers in terms of the Hessenberg determinants.

Explicit, determinantal, recursive formulas and relations of the Peters polynomials and numbers

“…Lemma 2.1 (Comtet 31,p. 134 and 139 and Guo et al 32 ). For n ≥ k ≥ 0, the Bell polynomials of the second kind B n, k (x 1 , x 2 , … , x n − k + 1 ) are defined by…”

“…In this section, we establish several recursive relations for the Peters polynomials and numbers, and Boole polynomials and numbers by applying the Hessenberg determinant. In the existing literature on the subject, one can find a fairly large number of papers relating the several types of determinants such as Hessenberg determinant and Hankel determinant with some special numbers and polynomials to algebra and combinatorial number theory, (see, for example, previous studies 25,32,[36][37][38][39][40][41][42][43] ). Proof.…”