Research Article
BibTex RIS Cite

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: 1

Abstract

References

  • [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

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: 1

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

References

  • [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.
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Takao Komatsu 0000-0001-6204-5368

José L. Ramírez This is me 0000-0002-8028-9312

Publication Date April 2, 2020
Published in Issue Year 2020 Volume: 49 Issue: 2

Cite

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. April 2020;49(2):565-577. doi:10.15672/hujms.568340
Chicago Komatsu, Takao, and José L. Ramírez. “Convolutions of the Bi-Periodic Fibonacci Numbers”. Hacettepe Journal of Mathematics and Statistics 49, no. 2 (April 2020): 565-77. https://doi.org/10.15672/hujms.568340.
EndNote Komatsu T, Ramírez JL (April 1, 2020) Convolutions of the bi-periodic Fibonacci numbers. Hacettepe Journal of Mathematics and Statistics 49 2 565–577.
IEEE T. Komatsu and J. L. Ramírez, “Convolutions of the bi-periodic Fibonacci numbers”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 2, pp. 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 (April 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 and José L. Ramírez. “Convolutions of the Bi-Periodic Fibonacci Numbers”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 2, 2020, pp. 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.

Cited By

Some identities involving the bi-periodic Fibonacci and Lucas polynomials
AIMS Mathematics
https://doi.org/10.3934/math.2023294

https://dergipark.org.tr/en/pub/hujms