Bi-Periodic Balancing Quaternions

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.

Keywords

• Bi-periodic balancing numbers
• Binet formula
• Generating function
• Cassini identity
• Catalan identity

References

1. Behera, A., Panda, G.K., {\em On the square roots of triangular numbers}, Fibonacci Quarterly, \textbf{37(2)}(1999), 98--105.
2. Edson, M., Yayenie, O., {\em A new generalization of Fibonacci sequences and extended Binet's formula}, Integers, \textbf{9(6)}(2009), 639--654.
3. Hal\i c\i , S., {\em On Fibonacci quaternions}, Advances in Applied Clifford Algebras, \textbf{22(2)}(2012), 321--327.
4. Hamilton, W.R., Lectures on quaternions, Dublin, 1853.
5. Horadam, A.F., {\em Complex Fibonacci numbers and Fibonacci quaternions}, The American Mathematical Monthly, \textbf{70(3)}(1963), 289--291.
6. Iyer, M.R., {\em Some results on Fibonacci quaternions}, Fibonacci Quarterly, \textbf{7(2)}(1969), 201--210.
7. Jordan, J.H., {\em Gaussian Fibonacci and Lucas numbers}, Fibonacci Quarterly, \textbf{3}(1965), 315--318.
8. Koshy, T., Fibonacci and Lucas numbers with applications, Wiley, New York, 2001.

9. 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.
10. 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.
11. 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.