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## Convolutions of the bi-periodic Fibonacci numbers

#### Takao KOMATSU [1] , José L. RAMÍREZ [2]

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.
bi-periodic Fibonacci numbers, convolutions, symmetric formulas
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Birincil Dil en Matematik Matematik Orcid: 0000-0001-6204-5368Yazar: Takao KOMATSU Kurum: Zhejiang Sci-Tech UniversityÜlke: China Orcid: 0000-0002-8028-9312Yazar: José L. RAMÍREZ Kurum: Universidad Nacional de ColombiaÜlke: Colombia Yayımlanma Tarihi : 2 Nisan 2020
 Bibtex @araştırma makalesi { hujms568340, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe Üniversitesi}, year = {2020}, volume = {49}, pages = {565 - 577}, doi = {10.15672/hujms.568340}, title = {Convolutions of the bi-periodic Fibonacci numbers}, key = {cite}, author = {KOMATSU, Takao and RAMÍREZ, José L.} } APA KOMATSU, T , RAMÍREZ, J . (2020). Convolutions of the bi-periodic Fibonacci numbers. Hacettepe Journal of Mathematics and Statistics , 49 (2) , 565-577 . DOI: 10.15672/hujms.568340 MLA KOMATSU, T , RAMÍREZ, J . "Convolutions of the bi-periodic Fibonacci numbers". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 565-577 Chicago KOMATSU, T , RAMÍREZ, J . "Convolutions of the bi-periodic Fibonacci numbers". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 565-577 RIS TY - JOUR T1 - Convolutions of the bi-periodic Fibonacci numbers AU - Takao KOMATSU , José L. RAMÍREZ Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.568340 DO - 10.15672/hujms.568340 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 565 EP - 577 VL - 49 IS - 2 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.568340 UR - https://doi.org/10.15672/hujms.568340 Y2 - 2019 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics Convolutions of the bi-periodic Fibonacci numbers %A Takao KOMATSU , José L. RAMÍREZ %T Convolutions of the bi-periodic Fibonacci numbers %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 2 %R doi: 10.15672/hujms.568340 %U 10.15672/hujms.568340 ISNAD KOMATSU, Takao , RAMÍREZ, José L. . "Convolutions of the bi-periodic Fibonacci numbers". Hacettepe Journal of Mathematics and Statistics 49 / 2 (Nisan 2020): 565-577 . https://doi.org/10.15672/hujms.568340 AMA KOMATSU T , RAMÍREZ J . Convolutions of the bi-periodic Fibonacci numbers. Hacettepe Journal of Mathematics and Statistics. 2020; 49(2): 565-577. Vancouver KOMATSU T , RAMÍREZ J . Convolutions of the bi-periodic Fibonacci numbers. Hacettepe Journal of Mathematics and Statistics. 2020; 49(2): 577-565.

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