Convolutions of the bi-periodic Fibonacci numbers

Abstract

Let q n be the bi-periodic Fibonacci numbers, defined by q n = c(n)q n−1 + q n−2 (n ≥ 2) with q 0 = 0 and q 1 = 1, where c(n) = a if n is even, c(n) = b if n is odd, where a and b are nonzero real numbers. When c(n) = a = b = 1, q n = F n are Fibonacci numbers. In this paper, the convolution identities of order 2, 3 and 4 for the bi-periodic Fibonacci numbers q n are given with binomial (or multinomial) coefficients, by using the symmetric formulas.

The relations between bi-periodic jacobsthal and bi-periodic jacobsthal lucas sequence

“…Gul studied bi-periodic Jacobsthal and Jacobsthal-Lucas quaternions in [11]. Komatsu, Ramírez studied on convolutions of the biperiodic Fibonacci numbers in [12].…”

A New Generalization of Jacobsthal Lucas Numbers (Bi-Periodic Jacobsthal Lucas Sequence)

“…He also found some interesting identities between the above two sequences. The authors in [8], [9], [10], [11], [12], [13], [14], [15] gave interesting properties of bi-periodic sequences.…”

“…In [18], Gul studied on bi-periodic Jacobsthal and Jacobsthal-Lucas quaternions. In [15], Komatsu and Ramírez, gave convolutions of the bi-periodic Fibonacci numbers. Now in this paper, just as the generalized Jacobsthal sequence and the others mentioned above, we define a new generalization for the Jacobsthal-Lucas sequence which we shall also call the biperiodic Jacobsthal-Lucas sequence.…”