A Note on Horadam Hybrinomials

Abstract

This paper ensures an extensive survey of the generalization of the various hybrid numbers and hybrid polynomials especially as part of its enhancing importance in the disciplines of mathematics and physics. In this paper, by using the Horadam polynomials, we define the Horadam hybrid polynomials called Horadam hybrinomials. We obtain some special cases and algebraic properties of the Horadam hybrinomials such as recurrence relation, generating function, exponential generating function, Binet formula, summation formulas, Catalan's identity, Cassini's identity and d'Ocagne's identity, respectively. Moreover, we give some applications related to the Horadam hybrinomials in matrices.

Keywords

• Horadam number
• Horadam polynomial
• Hybrid number

References

1. [1] A. F. Horadam, Basic properties of a certain generalized sequence of numbers, Fibonacci Q., 3 (1965), 161-176.
2. [2] A. F. Horadam, Generating functions for powers of a certain generalized sequence of numbers, Duke Math. J., 32 (1965), 437-446, .
3. [3] T. Koshy, Fibonacci and Lucas Numbers with Applications, Vol. 1, Second edition, Pure and Applied Mathematics (Hoboken), John Wiley & Sons, Inc., Hoboken, NJ, 2018.
4. [4] S. Falcon, A. Plaza, On the Fibonacci k-numbers, Chaos Solitons Fractals, 32 (5) (2007), 1615-1624.
5. [5] E. G. Kocer, N. Tuglu, A. Stakhov, On the m-extension of the Fibonacci and Lucas p􀀀numbers, Chaos Solitons Fractals, 40 (4) (2009), 1890-1906.
6. [6] C. Kızılateş, New families of Horadam numbers associated with finite operators and their applications, Math. Meth. Appl. Sci., 44 (2021), 14371-14381.
7. [7] T. Horzum, E. G. Kocer, On some properties of Horadam polynomials, Int. Math. Forum., 25 (4) (2009), 1243-1252.
8. [8] M. Özdemir, Introduction to hybrid numbers, Adv. Appl. Clifford Algebras, 28 (1) (2018).

9. [9] A. Szynal-Liana, I. Wloch, The Fibonacci hybrid numbers, Utilitas Math., 110 (2019), 3-10.
10. [10] A. Szynal-Liana, I. Wloch, On Jacosthal and Jacosthal-Lucas hybrid numbers, Ann. Math. Sil., 33 (1) (2019), 276-283.
11. [11] A. Szynal-Liana, I. Wloch, On Pell and Pell–Lucas Hybrid Numbers, Commentat. Math., 58 (2018), 11-17.
12. [12] A. Szynal-Liana, The Horadam hybrid numbers, Discuss. Math. Gen. Algebra Appl., 38 (1) (2018), 91-98.
13. [13] C. Kızılateş, A new generalization of Fibonacci hybrid and Lucas hybrid numbers, Chaos Solitons Fractals, 130 (2020), 1-5.
14. [14] P. Catarino, On k-Pell hybrid numbers, J. Discrete Math. Sci. Cryptography, 22 (1) (2019), 83-89.
15. [15] G. Cerda Moreles, Investigation of generalized hybrid Fibonacci numbers and their properties, Appl. Math. E-Notes, 21 (2021), 110-118.
16. [16] G. Cerda Moreles, Introduction to third-order Jacobsthal and modified third-order Jacobsthal hybrinomials, Discuss. Math. Gen. Algebra Appl., 41 (1) (2021), 139-152.
17. [17] Y. Alp, E. G. Kocer, Hybrid Leonardo numbers, Chaos Solitons Fractals, 150 (2021), 1-5.
18. [18] D. Tasci, E. Sevgi, Some properties between Mersenne, Jacobsthal and Jacobsthal-Lucas hybrid numbers, Chaos Solitons Fractals, 146 (2021), 1-4.
19. [19] E. Tan, NR. Ait-Amrane, On a new generalization of Fibonacci hybrid numbers, (2020), arXiv:2006.09727.
20. [20] S. Petroudi, M. Pirouz, A. Özkoç, The Narayana polynomial and Narayana hybrinomial sequences, Konuralp J. Math., 9 (1) (2021), 90-99.
21. [21] A. Özkoç, A new generalization of Tribonacci hybrinomial, Bull. Int. Math. Virtual Inst., 11 (3) (2021), 555-568.
22. [22] R. Vieira, M. Mangueira, F. R. Alves, P. M. M. Cruz Catarino, Padovan and Perrin Hybrid Number Identities, Commun. Adv. Math. Sci., 4 (4) (2021), 190-197.
23. [23] E. Polatlı, A note on ratios of Fibonacci hybrid and Lucas hybrid numbers, Notes Number Theory Discrete Math., 27 (3) (2021), 73-78.
24. [24] A. Szynal-Liana, I. Wloch, Introduction to Fibonacci and Lucas hybrinomials, Complex Var. Elliptic Eq., 65 (10) (2020), 1736-1747.
25. [25] M. Liana, A. Szynal-Liana, I. Wloch, On Pell hyrinomials, Miskolc Math. Notes, 20 (2019), 1051-1062.
26. [26] S. Falcon, On the generating matrices of the k-Fibonacci numbers, Proyecciones J. Math., 32 (4) (2013), 347-357.
27. [27] P. Catarino, A note on certain matrices with h(x)-Fibonacci quaternion polynomials, J. Differ. Eq. Appl., 22 (2016), 343-351.
28. [28] C. Kızılateş, P. Catarino, N. Tuglu, On the bicomplex generalized Tribonacci quaternions, Mathematics, 7 (1) (2019), 80.
29. [29] C. Kızılateş, A Note on Horadam hybrinomials, (2020), Preprints:2020010116.