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 $1 \leq s \leq r$, 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.