Theory and Applications of the Conditional Type Hyperbolic Functions

Cahit Köme; Zeynep Demirel

doi:10.31801/cfsuasmas.1622874

Theory and Applications of the Conditional Type Hyperbolic Functions

Abstract

In this paper, we introduce conditional and symmetric conditional hyperbolic $\textit{sine}$ and $\textit{cosine}$ functions by replacing a discrete variable with a continuous one and employing conditionally defined sequences. We establish fundamental relations between these functions and derive their generating and exponential generating functions, recurrence relations, sums and differences, double arguments, derivatives, De Moivre-type identities, and several key properties. Furthermore, we construct a quasi-sine conditional hyperbolic function by selecting an appropriate trigonometric function. Using these results, we develop three-dimensional conditional spiral and shofar structures. Finally, we present numerical and graphical illustrations to support the theoretical findings.

Keywords

References

Ahmed, S., Shah, K., Jahan, S., and Abdeljawad, T. (2023). An efficient method for the fractional electric circuits based on Fibonacci wavelet. Results in Physics, 52:106753.
Ait-Amrane, N. R. and Belbachir, H. (2022). Bi–periodic $r$–Fibonacci sequence and bi-periodic $r$–Lucas sequence of type $s$. Hacettepe Journal of Mathematics and Statistics, 51(3):680–699.
Biban, G., Chugh, R., and Panwar, A. (2023). Image encryption based on 8D hyperchaotic system using Fibonacci Q–matrix. Chaos, Solitons & Fractals, 170:113396.
Bilgici, G. (2014). Two generalizations of Lucas sequence. Applied Mathematics and Computation, 245:526–538.
Daşdemir, A., Şentürk, T. D., and Ünal, Z. (2020). On recursive hyperbolic functions in Fibonacci–Lucas sense. Hacettepe Journal of Mathematics and Statistics, 49(6):2046–2062.
Edson, M., Lewis, S., and Yayenie, O. (2011). The $k$−periodic Fibonacci sequence and an extended Binet’s formula. Integers, 11:739–751.
Edson, M. and Yayenie, O. (2009). A new generalization of Fibonacci sequence & extended Binet’s formula. Integers, 9:639–654.
Falcon, S. and Plaza, Á. (2007). The $k$−Fibonacci sequence and the Pascal $2-$triangle. Chaos, Solitons & Fractals, 33(1):38–49.

Falcon, S. and Plaza, A. (2008a). The $k$−Fibonacci hyperbolic functions. Chaos, Solitons & Fractals, 38(2):409–420.
Falcon, S. and Plaza, A. (2008b). On the $3$−dimensional $k$−Fibonacci spirals. Chaos, Solitons & Fractals, 38(4):993–1003.
Gaffar, A., Joshi, A. B., Singh, S., and Srivastava, K. (2022). A high capacity multi–image steganog- raphy technique based on golden ratio and non–subsampled contourlet transform. Multimedia Tools and Applications, 81(17):24449–24476.
Koçer, E. G., Tuğlu, N., and Stakhov, A. (2008). Hyperbolic functions with second order recurrence sequences. Ars Combinatoria, 88:65–81.
Komornik, V., Loreti, P., and Pedicini, M. (2024). A quasi–ergodic approach to non–integer base expansions. Journal of Number Theory, 254:146–168.
Koshy, T. (2019). Fibonacci and Lucas Numbers with Applications. John Wiley & Sons.
Liang, Z., Qin, Q., and Zhou, C. (2022). An image encryption algorithm based on Fibonacci Q–matrix and genetic algorithm. Neural Computing and Applications, 34(21):19313–19341.
Oyewole, O. K., Abass, H. A., and Ogunsola, O. (2024). An improved subgradient extragradient self– adaptive algorithm based on the golden ratio technique for variational inequality problems in Banach spaces. Journal of Computational and Applied Mathematics, page 116420.
Sacco, R. G. and Torday, J. S. (2023). Systems biology of human aging: A Fibonacci time series model. Progress in Biophysics and Molecular Biology, 177:24–33.
Stakhov, A. and Aranson, S. (2011). Hyperbolic Fibonacci and Lucas functions,“Golden” Fibonacci goniometry, Bodnar’s geometry, and Hilbert’s fourth problem–Part I. Hyperbolic Fibonacci and Lucas functions and “Golden” Fibonacci goniometry. Applied Mathematics, 2(1):74–84.
Stakhov, A. and Rozin, B. (2005a). The golden shofar. Chaos, Solitons & Fractals, 26(3):677–684.
Stakhov, A. and Rozin, B. (2005b). On a new class of hyperbolic functions. Chaos, Solitons & Fractals, 23(2):379–389.
Stakhov, A. and Rozin, B. (2006). The continuous functions for the Fibonacci and Lucas $p$−numbers. Chaos, Solitons & Fractals, 28(4):1014–1025.
Stakhov, A. and Rozin, B. (2007). The “golden” hyperbolic models of universe. Chaos, Solitons & Fractals, 34(2):159–171.
Stakhov, A. and Tkachenko, I. (1993). Hyperbolic Fibonacci trigonometry. Reports of the Ukrainian Academy of Sciences, 208(7):9–14.
Stakhov, A. P. (2003). Hyperbolic Fibonacci and Lucas functions: A new mathematics for the alive nature. ITI.
Tan, E. and Leung, H. H. (2020). A note on congruence properties of the generalized bi–periodic Horadam sequence. Hacettepe Journal of Mathematics and Statistics, 49(6):2084–2093.
Venunath, M., Sujatha, P., and Koti, P. (2024). Identification of influential users in social media network using golden ratio optimization method. Soft Computing, 28(3):2207–2222.
Yazlik, Y., Köme, C., and Madhusudanan, V. (2018). A new generalization of Fibonacci and Lucas $p$−numbers. Journal of Computational Analysis and Applications, 25(4):657–669.
Zhou, T., Shen, J., Li, X., Wang, C., and Tan, H. (2020). Logarithmic encryption scheme for cyberphysical systems employing Fibonacci Q-matrix. Future Generation Computer Systems, 108:1307–1313.

Details

Primary Language

English

Subjects

Mathematical Methods and Special Functions

Journal Section

Research Article

Authors

Cahit Köme ^*
0000-0002-6488-9035
Türkiye

Zeynep Demirel
0009-0007-9699-8133
Türkiye

Publication Date

September 26, 2026

Submission Date

January 18, 2025

Acceptance Date

December 23, 2025

Published in Issue

Year 2026 Volume: 75 Number: 3

DOI

https://doi.org/10.31801/cfsuasmas.1622874

IZ

https://izlik.org/JA37WE28RT

Cite

RIS / Bibtex

APA

Köme, C., & Demirel, Z. (2026). Theory and Applications of the Conditional Type Hyperbolic Functions. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 75(3), 433-452. https://doi.org/10.31801/cfsuasmas.1622874

AMA

1.Köme C, Demirel Z. Theory and Applications of the Conditional Type Hyperbolic Functions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2026;75(3):433-452. doi:10.31801/cfsuasmas.1622874

Chicago

Köme, Cahit, and Zeynep Demirel. 2026. “Theory and Applications of the Conditional Type Hyperbolic Functions”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 75 (3): 433-52. https://doi.org/10.31801/cfsuasmas.1622874.

EndNote

Köme C, Demirel Z (September 1, 2026) Theory and Applications of the Conditional Type Hyperbolic Functions. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 75 3 433–452.

IEEE

[1]C. Köme and Z. Demirel, “Theory and Applications of the Conditional Type Hyperbolic Functions”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 75, no. 3, pp. 433–452, Sept. 2026, doi: 10.31801/cfsuasmas.1622874.

ISNAD

Köme, Cahit - Demirel, Zeynep. “Theory and Applications of the Conditional Type Hyperbolic Functions”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 75/3 (September 1, 2026): 433-452. https://doi.org/10.31801/cfsuasmas.1622874.

JAMA

1.Köme C, Demirel Z. Theory and Applications of the Conditional Type Hyperbolic Functions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2026;75:433–452.

MLA

Köme, Cahit, and Zeynep Demirel. “Theory and Applications of the Conditional Type Hyperbolic Functions”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 75, no. 3, Sept. 2026, pp. 433-52, doi:10.31801/cfsuasmas.1622874.

Vancouver

1.Cahit Köme, Zeynep Demirel. Theory and Applications of the Conditional Type Hyperbolic Functions. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2026 Sep. 1;75(3):433-52. doi:10.31801/cfsuasmas.1622874