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Year 2023, Volume: 15 Issue: 1, 192 - 202, 30.06.2023

Fatih Yılmaz Aybüke Ertaş

https://doi.org/10.47000/tjmcs.1133973

Abstract

References

  • Akbıyık, M., Yamaç Akbıyık, S., Yılmaz, F., The matrices of Pauli quaternions, their De Moivre’s and Euler’s formulas. International Journal of Geometric Methods in Modern Physics, 19(11)(2022), 2250175-610.
  • Alp, Y., Kocer, E.G., Some properties of Leonardo numbers, Konuralp Journal of Mathematics, 9(1)(2021), 183–189.
  • Alves, F.R.V., Seqüencia de Oresme e algumas propriedades (matriciais) generalizadas, (2019), 40-44.
  • Arslan, H., Gaussian Pell and Gaussian Pell-Lucas uaternions, Filomat, 35(5)(2021), 1609-1617.
  • Bozkurt, Ş.B., Yılmaz, F., Bozkurt, D., On the complex factorization of the Lucas sequence, Applied Mathematics Letters, 24(8)(2011), 1317- 1321.
  • Bozkurt, D., Da Fonseca, C.M., Yılmaz, F., The determinants of circulant and skew-circulant matrices with tribonacci numbers, Mathematical Sciences and Applications E-notes, 2(2)(2014), 67-75.
  • Catarino, P., Borges, A., On Leonardo numbers, Acta Math. Univ. Comen., 89(2019), 75–86.
  • Cook, C.K., Some sums related to sums of Oresme numbers, In Applications of Fibonacci Numbers, Springer, Dordrecht, (2004), 87-99.
  • Dağdeviren, A., Yüce, S., Dual quaternions and dual quaternionic curves, Filomat, 33(2019), 1037–1046.
  • Dağdeviren, A., Kürüz, F., On the Horadam hybrid quaternions, Advanced Studies: Euro-Tbilisi Mathematical Journal, 10(2022), 293–306.
  • Goy, T., Zatorsky, R., On Oresme numbers and their connection with Fibonacci and Pell numbers, Fibonacci Quart., 57(3)(2019), 238-245.
  • Halıcı, S., On complex Fibonacci Quaternions, Adv. Appl. Clifford Algebras, 23(2013), 105-112.
  • Horadam, A.F., The Generalized Fibonacci sequences, The American Math. Monthly, 68(5)(1961), 455-459.
  • Horadam, A.F., Basic properties of a certain generalized sequence of numbers, Fibonacci Quart, 3(3)(1965), 161–176.
  • Horadam, A.F., Pell identities, Fibonacci Quart., 9(3)(1971), 245-263.
  • Horadam, A.F., Oresme numbers, The Fibonacci Quarterly, 12(3)(1974), 267-271.
  • Horadam, A.F., Jacobsthal and Pell curves, The Fibonacci Quarterly, 26(1)(1988), 79-83.
  • Horadam, A.F., Jacobsthal representation numbers, The Fib. Quart, 34 (1994), 40-54.
  • Karaca, E., Yılmaz, F., Çalışkan, M., A unified approach: split quaternions with quaternion coeffcients and quaternions with dual coeffcients., Mathematics, 8(12)(2020), 2149.
  • Karaca, E., Yılmaz, F., An introduction to harmonic complex numbers and harmonic hybrid Fibonacci numbers : A unified approach, Notes Number Theory Discret. Math., 3(2022), 542-557.
  • Karatas¸, A., Halıcı, S., Horadam octonions , Analele Universitatii ”Ovidius” Constanta - Seria Matematica, 25(3)(2017), 97-106.
  • Koshy,T., Fibonacci and Lucas Numbers with Applications, A Wiley Interscience Publication, 2001.
  • Kürüz, F., Dağdeviren, A., Catarino, P., On Leonardo Pisano hybrinomials, Mathematics, 9(22)(2021), 2923.
  • Mangueira, M.C. dos S., Vieira, R.P.M., Alves, F.R.V., Catarino, P. M.M.C., The Oresme sequence: The generalization of its matrix form and its hybridization process, NNTDM, 27(1)(2021), 101-111.
  • Pethe, S., Horadam, A.F., Generalized Gaussian Fibonacci numbers, Bull. Austral. Math. Soc., 33(1986), 37-48.
  • Szynal-Liana, A. Wloch, I., Oresme hybrid numbers and hybrationals, Kragujevac Journal of Mathematics, 48(5)(2024), 747–753.
  • Ward, J.P., Quaternions and Cayley numbers: Algebra and Applications, Mathematics and Its Applications, Kluwer, Dordrecht, 403(1997).
  • Yılmaz, F., Bozkurt, D., The adjacency matrix of one type of directed graph and the Jacobsthal numbers and their determinantal representation, Journal of Applied Mathematics, (2012), 1-14.
  • Yılmaz, F., Bozkurt, D., On the Fibonacci and Lucas numbers, their sums and permanents of one type of Hessenberg matrices, Hacettepe Journal of Mathematics and Statistics, 43(6)(2014), 1001-1007.
  • Yılmaz, F., Sogabe T., A note on symetric k-tridiagonal matrix family and the Fibonacci numbers, Int. J. Pure and Appl. Math., 96(2)(2014), 289-298.
  • Yılmaz, F., Özkan, M., On the generalized Gaussian fibonacci numbers and Horadam hybrid numbers: A unified approach, Axioms, 11(6)(2022), 255.
  • Zhang, F., Quaternions and matrices of quaternions, Linear Algebra and Its Applications, 251(1997), 21–57.

On Quaternions with Gaussian Oresme Coefficients

Year 2023, Volume: 15 Issue: 1, 192 - 202, 30.06.2023

Fatih Yılmaz Aybüke Ertaş

https://doi.org/10.47000/tjmcs.1133973

Abstract

At this paper, we describe Gaussian Oresme numbers taking into account the Oresme numbers. Furthermore, we investigate their some basic characteristic properties such as Binet formula and Cassini identity, etc. Moreover, we define quaternions with Gaussian Oresme coefficients and obtain their some spectacular properties.

Keywords

Generating function Binet formula Cassini identity Catalan identity D’Ocagne’s identity

References

  • Akbıyık, M., Yamaç Akbıyık, S., Yılmaz, F., The matrices of Pauli quaternions, their De Moivre’s and Euler’s formulas. International Journal of Geometric Methods in Modern Physics, 19(11)(2022), 2250175-610.
  • Alp, Y., Kocer, E.G., Some properties of Leonardo numbers, Konuralp Journal of Mathematics, 9(1)(2021), 183–189.
  • Alves, F.R.V., Seqüencia de Oresme e algumas propriedades (matriciais) generalizadas, (2019), 40-44.
  • Arslan, H., Gaussian Pell and Gaussian Pell-Lucas uaternions, Filomat, 35(5)(2021), 1609-1617.
  • Bozkurt, Ş.B., Yılmaz, F., Bozkurt, D., On the complex factorization of the Lucas sequence, Applied Mathematics Letters, 24(8)(2011), 1317- 1321.
  • Bozkurt, D., Da Fonseca, C.M., Yılmaz, F., The determinants of circulant and skew-circulant matrices with tribonacci numbers, Mathematical Sciences and Applications E-notes, 2(2)(2014), 67-75.
  • Catarino, P., Borges, A., On Leonardo numbers, Acta Math. Univ. Comen., 89(2019), 75–86.
  • Cook, C.K., Some sums related to sums of Oresme numbers, In Applications of Fibonacci Numbers, Springer, Dordrecht, (2004), 87-99.
  • Dağdeviren, A., Yüce, S., Dual quaternions and dual quaternionic curves, Filomat, 33(2019), 1037–1046.
  • Dağdeviren, A., Kürüz, F., On the Horadam hybrid quaternions, Advanced Studies: Euro-Tbilisi Mathematical Journal, 10(2022), 293–306.
  • Goy, T., Zatorsky, R., On Oresme numbers and their connection with Fibonacci and Pell numbers, Fibonacci Quart., 57(3)(2019), 238-245.
  • Halıcı, S., On complex Fibonacci Quaternions, Adv. Appl. Clifford Algebras, 23(2013), 105-112.
  • Horadam, A.F., The Generalized Fibonacci sequences, The American Math. Monthly, 68(5)(1961), 455-459.
  • Horadam, A.F., Basic properties of a certain generalized sequence of numbers, Fibonacci Quart, 3(3)(1965), 161–176.
  • Horadam, A.F., Pell identities, Fibonacci Quart., 9(3)(1971), 245-263.
  • Horadam, A.F., Oresme numbers, The Fibonacci Quarterly, 12(3)(1974), 267-271.
  • Horadam, A.F., Jacobsthal and Pell curves, The Fibonacci Quarterly, 26(1)(1988), 79-83.
  • Horadam, A.F., Jacobsthal representation numbers, The Fib. Quart, 34 (1994), 40-54.
  • Karaca, E., Yılmaz, F., Çalışkan, M., A unified approach: split quaternions with quaternion coeffcients and quaternions with dual coeffcients., Mathematics, 8(12)(2020), 2149.
  • Karaca, E., Yılmaz, F., An introduction to harmonic complex numbers and harmonic hybrid Fibonacci numbers : A unified approach, Notes Number Theory Discret. Math., 3(2022), 542-557.
  • Karatas¸, A., Halıcı, S., Horadam octonions , Analele Universitatii ”Ovidius” Constanta - Seria Matematica, 25(3)(2017), 97-106.
  • Koshy,T., Fibonacci and Lucas Numbers with Applications, A Wiley Interscience Publication, 2001.
  • Kürüz, F., Dağdeviren, A., Catarino, P., On Leonardo Pisano hybrinomials, Mathematics, 9(22)(2021), 2923.
  • Mangueira, M.C. dos S., Vieira, R.P.M., Alves, F.R.V., Catarino, P. M.M.C., The Oresme sequence: The generalization of its matrix form and its hybridization process, NNTDM, 27(1)(2021), 101-111.
  • Pethe, S., Horadam, A.F., Generalized Gaussian Fibonacci numbers, Bull. Austral. Math. Soc., 33(1986), 37-48.
  • Szynal-Liana, A. Wloch, I., Oresme hybrid numbers and hybrationals, Kragujevac Journal of Mathematics, 48(5)(2024), 747–753.
  • Ward, J.P., Quaternions and Cayley numbers: Algebra and Applications, Mathematics and Its Applications, Kluwer, Dordrecht, 403(1997).
  • Yılmaz, F., Bozkurt, D., The adjacency matrix of one type of directed graph and the Jacobsthal numbers and their determinantal representation, Journal of Applied Mathematics, (2012), 1-14.
  • Yılmaz, F., Bozkurt, D., On the Fibonacci and Lucas numbers, their sums and permanents of one type of Hessenberg matrices, Hacettepe Journal of Mathematics and Statistics, 43(6)(2014), 1001-1007.
  • Yılmaz, F., Sogabe T., A note on symetric k-tridiagonal matrix family and the Fibonacci numbers, Int. J. Pure and Appl. Math., 96(2)(2014), 289-298.
  • Yılmaz, F., Özkan, M., On the generalized Gaussian fibonacci numbers and Horadam hybrid numbers: A unified approach, Axioms, 11(6)(2022), 255.
  • Zhang, F., Quaternions and matrices of quaternions, Linear Algebra and Its Applications, 251(1997), 21–57.
There are 32 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Fatih Yılmaz 0000-0001-7873-1979

Aybüke Ertaş 0000-0002-6133-5520

Publication Date June 30, 2023
Published in Issue Year 2023 Volume: 15 Issue: 1

Cite

APA Yılmaz, F., & Ertaş, A. (2023). On Quaternions with Gaussian Oresme Coefficients. Turkish Journal of Mathematics and Computer Science, 15(1), 192-202. https://doi.org/10.47000/tjmcs.1133973
AMA Yılmaz F, Ertaş A. On Quaternions with Gaussian Oresme Coefficients. TJMCS. June 2023;15(1):192-202. doi:10.47000/tjmcs.1133973
Chicago Yılmaz, Fatih, and Aybüke Ertaş. “On Quaternions With Gaussian Oresme Coefficients”. Turkish Journal of Mathematics and Computer Science 15, no. 1 (June 2023): 192-202. https://doi.org/10.47000/tjmcs.1133973.
EndNote Yılmaz F, Ertaş A (June 1, 2023) On Quaternions with Gaussian Oresme Coefficients. Turkish Journal of Mathematics and Computer Science 15 1 192–202.
IEEE F. Yılmaz and A. Ertaş, “On Quaternions with Gaussian Oresme Coefficients”, TJMCS, vol. 15, no. 1, pp. 192–202, 2023, doi: 10.47000/tjmcs.1133973.
ISNAD Yılmaz, Fatih - Ertaş, Aybüke. “On Quaternions With Gaussian Oresme Coefficients”. Turkish Journal of Mathematics and Computer Science 15/1 (June 2023), 192-202. https://doi.org/10.47000/tjmcs.1133973.
JAMA Yılmaz F, Ertaş A. On Quaternions with Gaussian Oresme Coefficients. TJMCS. 2023;15:192–202.
MLA Yılmaz, Fatih and Aybüke Ertaş. “On Quaternions With Gaussian Oresme Coefficients”. Turkish Journal of Mathematics and Computer Science, vol. 15, no. 1, 2023, pp. 192-0, doi:10.47000/tjmcs.1133973.
Vancouver Yılmaz F, Ertaş A. On Quaternions with Gaussian Oresme Coefficients. TJMCS. 2023;15(1):192-20.