Bi-periodic $r$-Fibonacci sequence and bi-periodic $r$-Lucas sequence of type $s$

Abstract

In the present paper, for a positive integer r, we study bi-periodic r-Fibonacci sequence and its family of companion sequences, bi-periodic r-Lucas sequence of type s with 1 ≤ s ≤ r, which extend the classical Fibonacci and Lucas sequences. Afterwards, we establish the link between the bi-periodic r-Fibonacci sequence and its companion sequences. Furthermore, we give their properties as linear recurrence relations, generating functions, explicit formulas and Binet forms.

Generalized bivariate conditional Fibonacci and Lucas hybrinomials

“…They also explored the bi-periodic r−Lucas sequence of type s, where s ranges from 1 to r, extending the classical Fibonacci and Lucas sequences. [1]. Belbachir and Bencherif [4] have generalized to bivariate polynomials of the Fibonacci and Lucas, properties obtained for Chebyshev polynomials.…”