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Bi-Periodic Balancing Quaternions

Year 2020, Volume: 12 Issue: 2, 68 - 75, 31.12.2020
https://doi.org/10.47000/tjmcs.701638

Abstract

In this paper, we first define the bi-periodic balancing numbers and quaternions. We give the generating function
and Binet formula for this quaternion. Then, we obtain some identities and properties including this quaternion.

References

  • Behera, A., Panda, G.K., {\em On the square roots of triangular numbers}, Fibonacci Quarterly, \textbf{37(2)}(1999), 98--105.
  • Edson, M., Yayenie, O., {\em A new generalization of Fibonacci sequences and extended Binet's formula}, Integers, \textbf{9(6)}(2009), 639--654.
  • Hal\i c\i , S., {\em On Fibonacci quaternions}, Advances in Applied Clifford Algebras, \textbf{22(2)}(2012), 321--327.
  • Hamilton, W.R., Lectures on quaternions, Dublin, 1853.
  • Horadam, A.F., {\em Complex Fibonacci numbers and Fibonacci quaternions}, The American Mathematical Monthly, \textbf{70(3)}(1963), 289--291.
  • Iyer, M.R., {\em Some results on Fibonacci quaternions}, Fibonacci Quarterly, \textbf{7(2)}(1969), 201--210.
  • Jordan, J.H., {\em Gaussian Fibonacci and Lucas numbers}, Fibonacci Quarterly, \textbf{3}(1965), 315--318.
  • Koshy, T., Fibonacci and Lucas numbers with applications, Wiley, New York, 2001.
  • Ozkan K\i z\i l\i rmak, G., Tasc\i , D., {\em Expression of reciprocal sum of Gaussian Lucas sequences by Lambert series,} Journal of Science and Arts, \textbf{48(3)}(2019), 587--592.
  • Tan, E., Y\i lmaz, S., Sahin, M., {\em A note on bi-periodic Fibonacci and Lucas quaternions}, Chaos, Solitions and Fractals, \textbf{85}(2016), 138--142.
  • Tasc\i , D., Ozkan K\i z\i l\i rmak, G., {\em On the periods of bi-periodic Fibonacci and bi-periodic Lucas numbers}, Discrete Dynamics in Nature and Society, (2016), 1--5.
Year 2020, Volume: 12 Issue: 2, 68 - 75, 31.12.2020
https://doi.org/10.47000/tjmcs.701638

Abstract

References

  • Behera, A., Panda, G.K., {\em On the square roots of triangular numbers}, Fibonacci Quarterly, \textbf{37(2)}(1999), 98--105.
  • Edson, M., Yayenie, O., {\em A new generalization of Fibonacci sequences and extended Binet's formula}, Integers, \textbf{9(6)}(2009), 639--654.
  • Hal\i c\i , S., {\em On Fibonacci quaternions}, Advances in Applied Clifford Algebras, \textbf{22(2)}(2012), 321--327.
  • Hamilton, W.R., Lectures on quaternions, Dublin, 1853.
  • Horadam, A.F., {\em Complex Fibonacci numbers and Fibonacci quaternions}, The American Mathematical Monthly, \textbf{70(3)}(1963), 289--291.
  • Iyer, M.R., {\em Some results on Fibonacci quaternions}, Fibonacci Quarterly, \textbf{7(2)}(1969), 201--210.
  • Jordan, J.H., {\em Gaussian Fibonacci and Lucas numbers}, Fibonacci Quarterly, \textbf{3}(1965), 315--318.
  • Koshy, T., Fibonacci and Lucas numbers with applications, Wiley, New York, 2001.
  • Ozkan K\i z\i l\i rmak, G., Tasc\i , D., {\em Expression of reciprocal sum of Gaussian Lucas sequences by Lambert series,} Journal of Science and Arts, \textbf{48(3)}(2019), 587--592.
  • Tan, E., Y\i lmaz, S., Sahin, M., {\em A note on bi-periodic Fibonacci and Lucas quaternions}, Chaos, Solitions and Fractals, \textbf{85}(2016), 138--142.
  • Tasc\i , D., Ozkan K\i z\i l\i rmak, G., {\em On the periods of bi-periodic Fibonacci and bi-periodic Lucas numbers}, Discrete Dynamics in Nature and Society, (2016), 1--5.
There are 11 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Emre Sevgi 0000-0003-2711-9880

Dursun Taşçı 0000-0001-8357-4875

Publication Date December 31, 2020
Published in Issue Year 2020 Volume: 12 Issue: 2

Cite

APA Sevgi, E., & Taşçı, D. (2020). Bi-Periodic Balancing Quaternions. Turkish Journal of Mathematics and Computer Science, 12(2), 68-75. https://doi.org/10.47000/tjmcs.701638
AMA Sevgi E, Taşçı D. Bi-Periodic Balancing Quaternions. TJMCS. December 2020;12(2):68-75. doi:10.47000/tjmcs.701638
Chicago Sevgi, Emre, and Dursun Taşçı. “Bi-Periodic Balancing Quaternions”. Turkish Journal of Mathematics and Computer Science 12, no. 2 (December 2020): 68-75. https://doi.org/10.47000/tjmcs.701638.
EndNote Sevgi E, Taşçı D (December 1, 2020) Bi-Periodic Balancing Quaternions. Turkish Journal of Mathematics and Computer Science 12 2 68–75.
IEEE E. Sevgi and D. Taşçı, “Bi-Periodic Balancing Quaternions”, TJMCS, vol. 12, no. 2, pp. 68–75, 2020, doi: 10.47000/tjmcs.701638.
ISNAD Sevgi, Emre - Taşçı, Dursun. “Bi-Periodic Balancing Quaternions”. Turkish Journal of Mathematics and Computer Science 12/2 (December 2020), 68-75. https://doi.org/10.47000/tjmcs.701638.
JAMA Sevgi E, Taşçı D. Bi-Periodic Balancing Quaternions. TJMCS. 2020;12:68–75.
MLA Sevgi, Emre and Dursun Taşçı. “Bi-Periodic Balancing Quaternions”. Turkish Journal of Mathematics and Computer Science, vol. 12, no. 2, 2020, pp. 68-75, doi:10.47000/tjmcs.701638.
Vancouver Sevgi E, Taşçı D. Bi-Periodic Balancing Quaternions. TJMCS. 2020;12(2):68-75.