Binomial Transforms for Hybrid Numbers Defined Through Fibonacci and Lucas Number Components

Abstract

The main purpose of covering this article has been to examine the hybrid numbers defined through Fibonacci and Lucas number components $\{\hat{H}% _{U,n}\}$ and $\{\hat{H}_{V,n}\}$~and their binomial transforms $\{b\hat{H}% _{U,n}\}$ and $\{b\hat{H}_{V,n}\},$ respectively. Firstly, general sum formulas, binomial identities, which are not in the literature yet, about hybrid numbers $\{\hat{H}_{U,n}\}$ and $\{\hat{H}_{V,n}\}$ are discussed. Then, the recurrence relation is obtained for $\{b\hat{H}_{U,n}\}$ and $\{b% \hat{H}_{V,n}\}$ and some results have been found for the new sequence.

Keywords

• Binomial transform
• hybrid numbers
• Lucas sequences of …rst and second kind.

References

1. [1] P. Ribenboim. My Numbers, My Friends, Popular Lectures on Number Theory. Springer-Verlag, New York, Inc. 2000.
2. [2] T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley and Sons, Volume 1, Second Edition, 2018.
3. [3] B. Demirtürk Bitim, N. Topal, Quaternions via Generalized Fibonacci and Lucas Number Components, Math. Reports 21(71), 2, (2019), 239-247.
4. [4] A. F. Horadam, Basic Properties of a Certain Generalized Sequence of Numbers, The Fibonacci Quart., 3(3), (1965), 161-176.
5. [5] A. F. Horadam, Generating Functions for Powers of a Certain Generalized Sequence of Numbers, Duke Math J., 32(3), (1965), 437-446.
6. [6] M. Özdemir, Introduction to Hybrid Numbers, Adv. Appl. Cli¤ord Algebr. 28:11, (2018), no. 1, Art. 11, 32 pp.
7. [7] A. S. Liana, I. Wloch, The Fibonacci Hybrid Numbers. Utilitas Math. (2019), 110: 3-10.
8. [8] A. S. Liana, I. Wloch, Jacobsthal and Jacobsthal Lucas Hybrid Numbers, Annales Mathematiccae Silesianae, (2019), pp. 276-283.

9. [9] A. S. Liana, The Horadam Hybrid Numbers. Discuss. Math. Gen. Algebra Appl. 38(1), (2018), 91-98.
10. [10] T. D. ¸Sentürk, G.Bilgici, A. Dasdemir, Z. Ünal, Study on Horadam Hybrid Numbers, Turk. J. Math. (2020), 44: 1212 -1221.
11. [11] P. Catarino, G. Bilgici, A Note on Modifed k-Pell Hybrid Numbers, Konuralp Journal of Mathematics, 8 (2) (2020), 229-233.
12. [12] E. Polatlı, Hybrid Numbers with Fibonacci and Lucas Hybrid Number Coefficients, preprint, (2020).
13. [13] Y. Taşyurdu, Tribonacci and Tribonacci-Lucas Hybrid Numbers, International Journal of Contemporary Mathematical Sciences 14(4), (2019), 245 - 254.
14. [14] H. Prodinger, Some Information about the Binomial Transform, The Fibonacci Quart., 32 (5), (1994), 412-415.
15. [15] R. L. Graham, D. E. Knuth, O. Patashnik, Concrete Mathematics, Addison Wesley Publishing Co., 1998.
16. [16] K. W. Chen, Identities from the Binomial Transform, J. Number Theory, (2007), 124, 142-150.
17. [17] S. Falcon, A. Plaza, Binomial Transforms of the k􀀀Fibonacci Sequences, Int. J. Nonlinear Sci. Numer. Simul., 10, (2009), 1305-1316.
18. [18] E. Polatlı, On Certain Properties of Quadrapell Sequences, Karaelmas Science and Engineering Journal, 8(1), (2018), 305-308.
19. [19] F. Kaplan, A. Özkoç Öztürk. On the Binomial Transforms of the Horadam Quaternion Sequence, Mathematical Methods in the Applied Sciences, DOI: 10.1002/mma.7325.
20. [20] K. N. Boyadzhiev, Notes on the Binomial Transform: Theory and Table with Appendix on Stirling Transform, World Scienti…c, Singapore, 2018.
21. [21] N. Yılmaz, Binomial Transforms of the Balancing and Lucas-Balancing Polynomials, Contributions to the Discrete Mathematics, 15(3), (2020), 133-144.
22. [22] S.K. Ghosal, J.K. Mandal, Binomial Transform Based Fragile Watermarking for Image Authentication, J. Inf. Secur. Appl., 19 (4-5) (2014), pp. 272-281.