Bi-Periodic (p,q)-Fibonacci and Bi-Periodic (p,q)-Lucas Sequences

Abstract

In this paper, we define bi-periodic (p,q)-Fibonacci and bi-periodic (p,q)-Lucas sequences, which generalize Fibonacci type, Lucas type, bi-periodic Fibonacci type and bi-periodic Lucas type sequences, using recurrence relations of (p,q)-Fibonacci and (p,q)-Lucas sequences. Generating functions and Binet formulas that allow us to calculate the nth terms of these sequences are given and the convergence properties of their consecutive terms are examined. Also, we prove some fundamental identities of bi-periodic (p,q)-Fibonacci and bi-periodic (p,q)-Lucas sequences conform to the well-known properties of Fibonacci and Lucas sequences.

Keywords

• bi-periodic fibonacci numbers
• fibonacci number
• generalized fibonacci numbers
• bi-periodic lucas numbers
• lucas number