- Hacettepe Journal of Mathematics and Statistics
- Vol: 49 Issue: 2
- Convolutions of the bi-periodic Fibonacci numbers
Convolutions of the bi-periodic Fibonacci numbers
Authors : Takao Komatsu, José L. Ramírez
Pages : 565-577
Doi:10.15672/hujms.568340
View : 12 | Download : 5
Publication Date : 2020-04-02
Article Type : Research
Abstract :Let $q_n$ be the bi-periodic Fibonacci numbers, defined by $q_n=c(n)q_{n-1}+q_{n-2}$ ($n\ge 2$) 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.Keywords : bi-periodic Fibonacci numbers, convolutions, symmetric formulas