THE HYBRID FORM OF THE HYPERBOLIC HORADAM MATRIX FUNCTIONS

Efruz Özlem Mersin

THE HYBRID FORM OF THE HYPERBOLIC HORADAM MATRIX FUNCTIONS

Abstract

In the present paper, a special type of hybrid number is introduced whose components are the hyperbolic Horadam sine and cosine matrix functions. These hybrid numbers and their symmetrical forms are explored in detail, and their recursive relations as well as hyperbolic properties are investigated. The paper also includes the Cassini, Catalan, and De Moivre identities, along with the Pythagorean Theorem, all for the hybrid form of the hyperbolic Horadam sine and cosine matrix functions. Furthermore, the hybrid forms of the quasi-sine Horadam matrix function and three-dimensional Horadam matrix spiral related to these matrix functions are presented, offering new insights into their mathematical properties.

Keywords

References

Al-Mohy, A. H., Higham, N. J., (2009), A new scaling and squaring algorithm for the matrix exponential, SIAM Journal on Matrix Analysis and Applications, 31(3), pp. 970-989. https://doi.org/10.1137/09074721X
Bahşi, M., Mersin, E. Ö., (2022), On the Hyperbolic Horadam Matrix Functions, Hacettepe Journal of Mathematics and Statistics, 51(6), pp. 1550-1562. https://doi.org/10.15672/hujms.1092305
Bahşi, M., Solak, S., (2019), Hyperbolic Horadam functions, Gazi University Journal of Science, 32(3), pp. 956-965. https://doi.org/10.35378/gujs.441422
Bahşi, M., Solak, S., (2018), On the hyperbolic Fibonacci matrix functions, TWMS Journal of Applied and Engineering Mathematics, 8(2), pp. 454-465.
Belbachir, H., Belkhir, A., (2018), On some generalizations of Horadam’s numbers, Filomat, 32(14), pp. 5037-5052. https://doi.org/10.2298/FIL1814037B
Defez, E., Sastre, J., Ib´a˜nez, J. J., Peinado, J., (2016), Solving engineering models using hyperbolic matrix functions, Applied Mathematical Modelling, 40(4), pp. 2837-2844. https://doi.org/10.1016/j.apm.2015.09.050
Defez, E., Sastre, J., Ib´a˜nez, J. J., Ruiz, P. A., (2009), Computing matrix functions solving coupled differential models, Mathematical and Computer Modelling, 50(5-6), pp. 831-839. https://doi.org/10.1016/j.mcm.2009.05.012
Defez, E., Sastre, J., Ib´a˜nez, J. J., Ruiz, P. A., (2013), Computing matrix functions arising in engineering models with orthogonal matrix polynomials, Mathematical and Computer Modelling, 57(7-8), pp. 1738-1743. https://doi.org/10.1016/j.mcm.2011.11.022

Defez, E., Jodar, L., (1998), Some applications of Hermite matrix polynomials series expansions, Journal of Computational and Applied Mathematics, 99(1-2), pp. 105-117. https://doi.org/10.1016/S0377- 0427(98)00149-6
Edson, M., Yayenie, O., (2009), A new generalization of Fibonacci sequences and extended Binet’s formula, Integers, 9(6), pp. 639–654. https://doi.org/10.1515/INTEG.2009.051
Falc´on, S., Plaza, ´A., (2007), On the Fibonacci k-numbers, Chaos, Solitons & Fractals, 32(5), pp. 1615-1624. https://doi.org/10.1016/j.chaos.2006.09.022
Hargreaves, G. I., Higham, N. J., (2005), Efficient algorithms for the matrix cosine and sine, Numerical Algorithms, 40(4), pp. 383-400. https://doi.org/10.1007/s11075-005-8141-0
He, T. X., Shiue, P. J. S., (2009), On sequences of numbers and polynomials defined by linear recurrence relations of order 2, International Journal of Mathematics & Mathematical Sciences, 709386. https://doi.org/10.1155/2009/709386.
Higham, N. J., (2008), Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA.
Horadam, A. F., (1965), Basic properties of a certain generalized sequence of numbers, The Fibonacci Quarterly, 3(3), pp. 161-176. https://doi.org/10.1080/00150517.1965.12431416
Jodar, L., Navarro, E., Posso, A., Casaban, M., (2003), Constructive solution of strongly coupled continuous hyperbolic mixed problems, Applied Numerical Mathematics, 47(3-4), pp. 477-492. https://doi.org/10.1016/S0168-9274(03)00073-4
Kızılateş, C., (2020), A new generalization of Fibonacci hybrid and Lucas hybrid numbers, Chaos, Solitons & Fractals, 130, 109449. https://doi.org/10.1016/j.chaos.2019.109449
Kızılateş, C., (2021), New families of Horadam numbers associated with finite operators and their applications, Mathematical Methods in the Applied Sciences, 44(18), pp. 14371-14381. https://doi.org/10.1002/mma.7702
Kızılateş, C., (2022), A note on Horadam hybrinomials, Fundamental Journal of Mathematics and Applications, 5(1), pp. 1-9. https://doi.org/10.33401/fujma.993546
Kızılateş, C., Kone, T., (2023), On special spacelike hybrid numbers with Fibonacci divisor number components, Indian Journal of Pure and Applied Mathematics, 54(1), 279-287. https://doi.org/10.1007/s13226-022-00252-7
Kılıç, E., Tan, E., (2009), More general identities involving the terms of W(a, b; p, q), Ars Combinatoria, 93, pp. 459-461.
Koshy, T., (2001), Fibonacci and Lucas numbers with applications, Wiley-Interscience, New York.
Mersin, E. Ö., (2023), Hyperbolic Horadam hybrid functions, Notes on Number Theory and Discrete Mathematics, 29(2), pp. 389-401. https://doi.org/10.7546/nntdm.2023.23.2.389-401
Moore, G., (2011), Orthogonal polynomial expansions for the matrix exponential, Linear Algebra and its Applications, 435(3), pp. 537-559. https://doi.org/10.1016/j.laa.2010.09.021
Özdemir, M., (2018), Introduction to hybrid numbers, Advances in Applied Clifford Algebras, 28(11). https://doi.org/10.1007/s00006-018-0833-3
Pozar, D., (1991), Microwave Engineering, Addison-Wesley, New York.
Sastre, J., Ib´a˜nez, J. J., Defez, E., Ruiz, P. A., (2011), Accurate matrix exponential computation to solve coupled differential models in engineering, Mathematical and Computer Modelling, 54(7-8), pp. 1835-1840. https://doi.org/10.1016/j.mcm.2010.12.049
Sastre, J., Ib´a˜nez, J. J., Ruiz, P. A., Defez, E., (2013), Efficient computation of the matrix cosine, Applied Mathematics and Computation, 219, pp. 7575-7585. https://doi.org/10.1016/j.amc.2013.01.043
Shi, Y., Ding, F., Chen, T., (2006), 2-Norm based recursive design of transmultiplexers with designable filter length, Circuits, Systems and Signal Processing, 25, pp. 447-462. https://doi.org/10.1007/s00034-004-1029-8
Shi, Y., Fang, H., (2010), Kalman filter based identification for systems with randomly missing measurements in a network environment, International Journal of Control, 83(3), pp. 538-551. https://doi.org/10.1080/00207170903273987
Shi, Y., Yu, B., (2009), Output feedback stabilization of networked control systems with random delays modeled by Markov chains, IEEE Transactions on Automatic Control, 54(7), pp. 1668-1674. https://doi.org/10.1109/TAC.2009.2020638
Soykan, Y., (2023), On generalized Fibonacci polynomials: Horadam polynomials, Earthline Journal of Mathematical Sciences, 11(1), pp. 23-114. https://doi.org/10.34198/ejms.11123.23114
Szynal-Liana, A., (2018), The Horadam hybrid numbers, Discussiones Mathematicae General Algebra and Applications, 38(1), pp. 91-98. https://doi.org/10.7151/dmgaa.1287
Szynal-Liana, A., Wloch, I., (2018), On Pell and Pell–Lucas Hybrid Numbers, Commentationes Mathematicae, 58(1-2), pp. 11-17.
Szynal-Liana, A., Wloch, I., (2019), On Jacopsthal and Jacopsthal-Lucas hybrid numbers, Annales Mathematicae Silesianae, 33(1), pp. 276-283. https://doi.org/10.2478/amsil-2018-0009
Szynal-Liana A., Wloch I., (2019), The Fibonacci hybrid numbers. Utilitas Math., 110, pp. 3–10.
Szynal-Liana, A., Wloch, I., (2020), Introduction to Fibonacci and Lucas hybrinomials, Complex Variables and Elliptic Equations, 65(10), pp. 1736-1747. https://doi.org/10.1080/17476933.2019.1681416
Şentürk, T. C., Bilgici, G., Daşdemir, A., Ünal, Z., (2020), A study on Horadam hybrid numbers, Turkish Journal of Mathematics, 44(4), pp. 1212-1221. https://doi.org/10.3906/mat-1908-77
Taşcı, D., Sevgi, E., (2021), Some properties between Mersenne, Jacobsthal and Jacobsthal-Lucas hybrid numbers, Chaos, Solitons and Fractals, 146, 110862. https://doi.org/10.1016/j.chaos.2021.110862
Yayenie, O., (2011), A note on generalized Fibonacci sequences, Applied Mathematics and Computation, 217(12), pp. 5603-5611. https://doi.org/10.1016/j.amc.2010.12.038
Yazlık, Y., Taşkara, N., (2012), A note on generalized k-Horadam sequence, Computers & Mathematics with Applications, 63(1), pp. 36-41. https://doi.org/10.1016/j.camwa.2011.10.055
Zhang, J.B., Ding, F., Shi, Y., (2009), Self-tuning control based on multi-innovation stochastic gradient parameter estimation, Systems and Control Letters, 58(1), pp. 69-75. https://doi.org/10.1016/j.sysconle.2008.08.005

Details

Primary Language

English

Subjects

Algebra and Number Theory

Journal Section

Research Article

Authors

Efruz Özlem Mersin ^*
0000-0001-6260-9063
Türkiye

Publication Date

October 1, 2025

Submission Date

November 21, 2024

Acceptance Date

February 24, 2025

Published in Issue

Year 2025 Volume: 15 Number: 10

IZ

https://izlik.org/JA72PE32NU

Cite

RIS / Bibtex

APA

Mersin, E. Ö. (2025). THE HYBRID FORM OF THE HYPERBOLIC HORADAM MATRIX FUNCTIONS. TWMS Journal of Applied and Engineering Mathematics, 15(10), 2505-2518. https://izlik.org/JA72PE32NU

AMA

1.Mersin EÖ. THE HYBRID FORM OF THE HYPERBOLIC HORADAM MATRIX FUNCTIONS. JAEM. 2025;15(10):2505-2518. https://izlik.org/JA72PE32NU

Chicago

Mersin, Efruz Özlem. 2025. “THE HYBRID FORM OF THE HYPERBOLIC HORADAM MATRIX FUNCTIONS”. TWMS Journal of Applied and Engineering Mathematics 15 (10): 2505-18. https://izlik.org/JA72PE32NU.

EndNote

Mersin EÖ (October 1, 2025) THE HYBRID FORM OF THE HYPERBOLIC HORADAM MATRIX FUNCTIONS. TWMS Journal of Applied and Engineering Mathematics 15 10 2505–2518.

IEEE

[1]E. Ö. Mersin, “THE HYBRID FORM OF THE HYPERBOLIC HORADAM MATRIX FUNCTIONS”, JAEM, vol. 15, no. 10, pp. 2505–2518, Oct. 2025, [Online]. Available: https://izlik.org/JA72PE32NU

ISNAD

Mersin, Efruz Özlem. “THE HYBRID FORM OF THE HYPERBOLIC HORADAM MATRIX FUNCTIONS”. TWMS Journal of Applied and Engineering Mathematics 15/10 (October 1, 2025): 2505-2518. https://izlik.org/JA72PE32NU.

JAMA

1.Mersin EÖ. THE HYBRID FORM OF THE HYPERBOLIC HORADAM MATRIX FUNCTIONS. JAEM. 2025;15:2505–2518.

MLA

Mersin, Efruz Özlem. “THE HYBRID FORM OF THE HYPERBOLIC HORADAM MATRIX FUNCTIONS”. TWMS Journal of Applied and Engineering Mathematics, vol. 15, no. 10, Oct. 2025, pp. 2505-18, https://izlik.org/JA72PE32NU.

Vancouver

1.Efruz Özlem Mersin. THE HYBRID FORM OF THE HYPERBOLIC HORADAM MATRIX FUNCTIONS. JAEM [Internet]. 2025 Oct. 1;15(10):2505-18. Available from: https://izlik.org/JA72PE32NU