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Araştırma Makalesi
BibTex RIS Kaynak Göster

Yıl 2024, Cilt: 8 Sayı: 2, 114 - 121, 30.12.2024

Adnan Karataş

https://doi.org/10.29002/asujse.1576933

Öz

Kaynakça

  • [1] Bayro-Corrochano E., (2021). A survey on quaternion algebra and geometric algebra applications in engineering and computer science 1995–2020. IEEE Access, 9: 104326–104355.
  • [2] Huang C., Li J., Gao G., (2023). Review of quaternion-based color image processing methods. Mathematics, 11: 2056.
  • [3] Salamin E., (1979). Application of quaternions to computation with rotations. Tech. rep., Working Paper.
  • [4] Horadam A.F., (1963). Complex Fibonacci numbers and Fibonacci quaternions. The American Mathematical Monthly, 70: 289–291.
  • [5] Iyer M.R., (1969). Some results on Fibonacci quaternions. The Fibonacci Quarterly, 7: 201–210.
  • [6] Halici S., (2012). On Fibonacci quaternions. Adv. Appl. Clifford Algebras, 22: 321–327.
  • [7] Polatli E., Kesim S., (2015). On quaternions with generalized Fibonacci and Lucas number components. Advances in Difference Equations, 1: 1–8.
  • [8] Ramirez J.L., (2015). Some combinatorial properties of the k-Fibonacci and the k-Lucas quaternions. Analele stiintifice ale Universitatii Ovidius Constanta. Seria Matematica, 23: 201–212.
  • [9] Akyigit M., Hüda Kösal H., Tosun M., (2014). Fibonacci generalized quaternions. Advances in Applied Clifford Algebras, 24: 631–641.
  • [10] Bitim B.D., (2019). Some identities of Fibonacci and Lucas quaternions by quaternion matrices. Düzce Üniversitesi Bilim ve Teknoloji Dergisi, 7: 606–615.
  • [11] Cerda-Morales G., (2017). Some properties of horadam quaternions. arXiv preprint, arXiv:1707.05918.
  • [12] Flaut C., Savin D., (2015). Quaternion algebras and generalized Fibonacci- Lucas quaternions. Advances in Applied Clifford Algebras, 25: 853–862.
  • [13] Halici S., Karataş A., (2017). Some matrix representations of Fibonacci quaternions and octonions. Advances in Applied Clifford Algebras, 27: 1233–1242.
  • [14] Irmak N., (2020). More identities for fibonacci and lucas quaternions. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 69: 369–375.
  • [15] Köme S., Köme C., Yazlık Y., (2019). Modified generalized Fibonacci and Lucas quaternions. Journal of Science and Arts, 19: 49-60.
  • [16] Patel B.K., Ray P.K., (2019). On the properties of (p, q)-Fibonacci and (p, q)-Lucas quaternions. Mathematical Reports, 21: 15–25.
  • [17] Polatlı E., (2016). A generalization of Fibonacci and Lucas quaternions. Advances in Applied Clifford Algebras, 26: 719–730.
  • [18] Swamy M., (1973). On generalized Fibonacci quaternions. The Fibonacci Quarterly, 11: 547–550.
  • [19] Tasci D., Buyukkose S., Kizilirmak G.O., Sevgi E., (2020). Properties of Lucas-sum graph. Journal of Science and Arts, 20: 313–316.
  • [20] Kilic E., Stakhov A., (2009). On the Fibonacci and Lucas p-numbers, their sums, families of bipartite graphs and permanents of certain matrices. Chaos, Solitons & Fractals, 40: 2210–2221.
  • [21] Koshy T., (2019), Fibonacci and Lucas Numbers with Applications. John Wiley & Sons.

Sum and Weighted Sum Formulas for Fibonacci and Lucas Quaternions

Yıl 2024, Cilt: 8 Sayı: 2, 114 - 121, 30.12.2024

Adnan Karataş

https://doi.org/10.29002/asujse.1576933

Öz

Fibonacci and Lucas quaternions are deeply studied in the literature after its definition by Horadam in 1963. Binet formula, generation function, Cassini, Catalan, Honsberger, and other identities for these sequences studied in different quaternion algebra types like split quaternion algebra, dual quaternion algebra, etc. Additionally, some of the generalizations for Fibonacci and Lucas quaternions are studied like k-Fibonacci quaternions, k-Lucas quaternions, Horadam quaternions, and more. However, despite being researched intensively, the summation formulas or weighted summation formulas for these sequences are not examined thoroughly. In this study, after briefly summarizing the literature, we focused on the summation and the weighted summation formulas for the Fibonacci and the Lucas quaternions. In addition, we provided generalized weighted summation formulas for the Fibonacci and Lucas quaternions. Moreover, we calculated generalized weighted summation formulas for the double coefficient Fibonacci and double coefficient Lucas quaternions which have two Fibonacci and Lucas coefficient in every unit of the quaternion, respectively.

Anahtar Kelimeler

Fibonacci Numbers Lucas Numbers Sum Formulas Weighted Sum Formulas Double Coefficient Fibonacci Numbers

Etik Beyan

The author declares that he has no conflict of interest.

Kaynakça

  • [1] Bayro-Corrochano E., (2021). A survey on quaternion algebra and geometric algebra applications in engineering and computer science 1995–2020. IEEE Access, 9: 104326–104355.
  • [2] Huang C., Li J., Gao G., (2023). Review of quaternion-based color image processing methods. Mathematics, 11: 2056.
  • [3] Salamin E., (1979). Application of quaternions to computation with rotations. Tech. rep., Working Paper.
  • [4] Horadam A.F., (1963). Complex Fibonacci numbers and Fibonacci quaternions. The American Mathematical Monthly, 70: 289–291.
  • [5] Iyer M.R., (1969). Some results on Fibonacci quaternions. The Fibonacci Quarterly, 7: 201–210.
  • [6] Halici S., (2012). On Fibonacci quaternions. Adv. Appl. Clifford Algebras, 22: 321–327.
  • [7] Polatli E., Kesim S., (2015). On quaternions with generalized Fibonacci and Lucas number components. Advances in Difference Equations, 1: 1–8.
  • [8] Ramirez J.L., (2015). Some combinatorial properties of the k-Fibonacci and the k-Lucas quaternions. Analele stiintifice ale Universitatii Ovidius Constanta. Seria Matematica, 23: 201–212.
  • [9] Akyigit M., Hüda Kösal H., Tosun M., (2014). Fibonacci generalized quaternions. Advances in Applied Clifford Algebras, 24: 631–641.
  • [10] Bitim B.D., (2019). Some identities of Fibonacci and Lucas quaternions by quaternion matrices. Düzce Üniversitesi Bilim ve Teknoloji Dergisi, 7: 606–615.
  • [11] Cerda-Morales G., (2017). Some properties of horadam quaternions. arXiv preprint, arXiv:1707.05918.
  • [12] Flaut C., Savin D., (2015). Quaternion algebras and generalized Fibonacci- Lucas quaternions. Advances in Applied Clifford Algebras, 25: 853–862.
  • [13] Halici S., Karataş A., (2017). Some matrix representations of Fibonacci quaternions and octonions. Advances in Applied Clifford Algebras, 27: 1233–1242.
  • [14] Irmak N., (2020). More identities for fibonacci and lucas quaternions. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 69: 369–375.
  • [15] Köme S., Köme C., Yazlık Y., (2019). Modified generalized Fibonacci and Lucas quaternions. Journal of Science and Arts, 19: 49-60.
  • [16] Patel B.K., Ray P.K., (2019). On the properties of (p, q)-Fibonacci and (p, q)-Lucas quaternions. Mathematical Reports, 21: 15–25.
  • [17] Polatlı E., (2016). A generalization of Fibonacci and Lucas quaternions. Advances in Applied Clifford Algebras, 26: 719–730.
  • [18] Swamy M., (1973). On generalized Fibonacci quaternions. The Fibonacci Quarterly, 11: 547–550.
  • [19] Tasci D., Buyukkose S., Kizilirmak G.O., Sevgi E., (2020). Properties of Lucas-sum graph. Journal of Science and Arts, 20: 313–316.
  • [20] Kilic E., Stakhov A., (2009). On the Fibonacci and Lucas p-numbers, their sums, families of bipartite graphs and permanents of certain matrices. Chaos, Solitons & Fractals, 40: 2210–2221.
  • [21] Koshy T., (2019), Fibonacci and Lucas Numbers with Applications. John Wiley & Sons.
Toplam 21 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Sayısal ve Hesaplamalı Matematik (Diğer)
Bölüm Araştırma Makalesi
Yazarlar

Adnan Karataş 0000-0003-3652-5354

Yayımlanma Tarihi 30 Aralık 2024
Gönderilme Tarihi 31 Ekim 2024
Kabul Tarihi 27 Aralık 2024
Yayımlandığı Sayı Yıl 2024 Cilt: 8 Sayı: 2

Kaynak Göster

APA Karataş, A. (2024). Sum and Weighted Sum Formulas for Fibonacci and Lucas Quaternions. Aksaray University Journal of Science and Engineering, 8(2), 114-121. https://doi.org/10.29002/asujse.1576933
 Kapak Resmi İndir
Aksaray J. Sci. Eng. | e-ISSN: 2587-1277 | Period: Biannually | Founded: 2017 | Publisher: Aksaray University | https://asujse.aksaray.edu.tr