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AI Summary & Key Takeaways
AI Insight⚡ TL;DR(Too Long; Didn't Read)
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.
📌 3 Key Takeaways
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Objective & Hypothesis: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.
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Key Findings & Data:When $c(n)=a=b=1$, $q_n=F_n$ are Fibonacci numbers.
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Conclusion & Impact: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.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:bi-periodic Fibonacci numbers · convolutions · symmetric formulas
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.
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Cite This Article
Komatsu, T., Ramírez, J. L. (2022). Convolutions of the bi-periodic Fibonacci numbers. Hacettepe Journal of Mathematics and Statistics, Cilt: 49 - Sayı: 2, 565–577. https://ajindex.com/articles/529678