Convolutions of the bi-periodic Fibonacci numbers

Year 2020, Volume: 49 Issue: 2, 565 - 577, 02.04.2020

Takao Komatsu , José L. Ramírez

https://doi.org/10.15672/hujms.568340

Cited By: 2

https://izlik.org/JA28HU44KN

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

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