KaydedilenlerDerginizi ekleyin

Dergiler / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 49 Issue: 2

Convolutions of the bi-periodic Fibonacci numbers

Takao Komatsu(Zhejiang Sci-Tech University)0000-0001-6204-5368
José L. Ramírez(Universidad Nacional de Colombia)0000-0002-8028-9312
Sayfa
—
Yayın Tarihi
2020-04-02
🏛️Bu makalenin bibliyografik verileri ve tam metin bağlantısı DergiPark üzerinden sağlanmıştır.
DergiPark Kaydı ↗
✨

Yapay Zeka Hızlı Bakış & Çıkarımlar

AI Insight
⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

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 Maddede Bu Makale

🎯
Amaç & Hipotez: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.
📊
Temel Bulgular & Veri:When $c(n)=a=b=1$, $q_n=F_n$ are Fibonacci numbers.
💡
Sonuç & Katkı: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.
🔬Tür:Research Article
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler: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.

🔖

Bu Makaleye Atıf Yap

Komatsu, T., Ramírez, J. L. (2020). Convolutions of the bi-periodic Fibonacci numbers. Hacettepe Journal of Mathematics and Statistics, Volume: 49 Issue: 2. https://doi.org/10.15672/hujms.568340

📄 Tam Metin PDF Okuyucu

📄

PDF görüntüleyici hazırlanıyor...

Kaynakça (20)

  1. T. Agoh and K. Dilcher, Convolution identities and lacunary recurrences for Bernoulli numbers, J. Number Theory, 124, 105–122, 2007.
  2. T. Agoh and K. Dilcher, Higher-order recurrences for Bernoulli numbers, J. Number Theory, 129, 1837–1847, 2009.
  3. T. Agoh and K. Dilcher, Higher-order convolutions for Bernoulli and Euler polyno- mials, J. Math. Anal. Appl. 419, 1235–1247, 2014.
  4. M. Alp, N. Irmak and L. Szalay, Two-Periodic ternary recurrences and their Binet- formula, Acta Math. Univ. Comenianae 2, 227–232, 2012.
  5. C. Cooper, Some identities involving differences of products of generalized Fibonacci numbers, Colloq. Math. 141 (1), 45–49, 2015.
  6. K. Dilcher and C. Vignat, General convolution identities for Bernoulli and Euler polynomials, J. Math. Anal. Appl. 435, 1478–1498, 2016.
  7. M. Edson and O. Yayenie, A new generalization of Fibonacci sequences and extended Binet’s Formula, Integers, 9 (A48), 639-654, 2009.
  8. N. Irmak and L. Szalay, On k-periodic binary recurrences, Ann. Math. Inform. 40, 25–35, 2012.
  9. T. Komatsu, Higher-order convolution identities for Cauchy numbers of the second kind, Proc. Jangjeon Math. Soc. 18, 369–383, 2015.
  10. T. Komatsu, Higher-order convolution identities for Cauchy numbers, Tokyo J. Math. 39, 225–239, 2016.
  11. T. Komatsu, Convolution identities for Tribonacci numbers, Ars Combin. 136, 199– 210, 2018.
  12. T. Komatsu and R. Li, Convolution identities for Tribonacci numbers with symmetric formulae, Math. Rep. (Bucur.) 21 (1), 27-47, 2019, arXiv:1610.02559.
  13. T. Komatsu, Z. Masakova and E. Pelantova, Higher-order identities for Fibonacci numbers, Fibonacci Quart. 52 (5), 150-163, 2014.
  14. T. Komatsu and G.K. Panda, On several kinds of sums involving balancing and Lucas- balancing numbers, Ars Combin. (to appear). arXiv:1608.05918.
  15. T. Komatsu and P.K. Ray, Higher-order identities for balancing numbers, arXiv:1608.05925, 2016.
  16. T. Komatsu and Y. Simsek, Third and higher order convolution identities for Cauchy numbers, Filomat 30, 1053–1060, 2016.
  17. R. Li, Convolution identities for Tetranacci numbers, arXiv:1609.05272.
  18. J.L. Ramírez, Bi-periodic incomplete Fibonacci sequences, Ann. Math. Inform. 42, 83–92, 2013.
  19. W. Wang, Some results on sums of products of Bernoulli polynomials and Euler polynomials, Ramanujan J. 32, 159–186, 2013.
  20. O. Yayenie, A note on generalized Fibonacci sequence, Applied. Math. Comp. 217 (12), 5603–5611, 2011.