Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2020, Cilt: 49 Sayı: 2, 565 - 577, 02.04.2020
https://doi.org/10.15672/hujms.568340

Öz

Kaynakça

  • [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.

Convolutions of the bi-periodic Fibonacci numbers

Yıl 2020, Cilt: 49 Sayı: 2, 565 - 577, 02.04.2020
https://doi.org/10.15672/hujms.568340

Öz

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.

Kaynakça

  • [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.
Toplam 20 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Matematik
Yazarlar

Takao Komatsu 0000-0001-6204-5368

José L. Ramírez Bu kişi benim 0000-0002-8028-9312

Yayımlanma Tarihi 2 Nisan 2020
Yayımlandığı Sayı Yıl 2020 Cilt: 49 Sayı: 2

Kaynak Göster

APA Komatsu, T., & Ramírez, J. L. (2020). Convolutions of the bi-periodic Fibonacci numbers. Hacettepe Journal of Mathematics and Statistics, 49(2), 565-577. https://doi.org/10.15672/hujms.568340
AMA Komatsu T, Ramírez JL. Convolutions of the bi-periodic Fibonacci numbers. Hacettepe Journal of Mathematics and Statistics. Nisan 2020;49(2):565-577. doi:10.15672/hujms.568340
Chicago Komatsu, Takao, ve José L. Ramírez. “Convolutions of the Bi-Periodic Fibonacci Numbers”. Hacettepe Journal of Mathematics and Statistics 49, sy. 2 (Nisan 2020): 565-77. https://doi.org/10.15672/hujms.568340.
EndNote Komatsu T, Ramírez JL (01 Nisan 2020) Convolutions of the bi-periodic Fibonacci numbers. Hacettepe Journal of Mathematics and Statistics 49 2 565–577.
IEEE T. Komatsu ve J. L. Ramírez, “Convolutions of the bi-periodic Fibonacci numbers”, Hacettepe Journal of Mathematics and Statistics, c. 49, sy. 2, ss. 565–577, 2020, doi: 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.
JAMA Komatsu T, Ramírez JL. Convolutions of the bi-periodic Fibonacci numbers. Hacettepe Journal of Mathematics and Statistics. 2020;49:565–577.
MLA Komatsu, Takao ve José L. Ramírez. “Convolutions of the Bi-Periodic Fibonacci Numbers”. Hacettepe Journal of Mathematics and Statistics, c. 49, sy. 2, 2020, ss. 565-77, doi:10.15672/hujms.568340.
Vancouver Komatsu T, Ramírez JL. Convolutions of the bi-periodic Fibonacci numbers. Hacettepe Journal of Mathematics and Statistics. 2020;49(2):565-77.