Convolutions of the bi-periodic Fibonacci numbers

Authors : Takao KOMATSU, José L. RAMÍREZ

Pages : 565-577

Doi:10.15672/hujms.568340

View : 13 | Download : 5

Publication Date : 2020-04-02

Article Type : Research Paper

Abstract :Let $q_n$ be the bi-periodic Fibonacci numbers, defined by $q_n=cinsert ignore into journalissuearticles values(n);q_{n-1}+q_{n-2}$ insert ignore into journalissuearticles values($n\ge 2$); with $q_0=0$ and $q_1=1$, where $cinsert ignore into journalissuearticles values(n);=a$ if $n$ is even, $cinsert ignore into journalissuearticles values(n);=b$ if $n$ is odd, where $a$ and $b$ are nonzero real numbers. When $cinsert ignore into journalissuearticles values(n);=a=b=1$, $q_n=F_n$ 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 insert ignore into journalissuearticles values(or multinomial); coefficients, by using the symmetric formulas.
Keywords : bi periodic Fibonacci numbers, convolutions, symmetric formulas