Hyperbolic Horadam Functions
Year 2019,
, 956 - 965, 01.09.2019
Mustafa Bahsı
,
Suleyman Solak
Abstract
This article introduce hyperbolic functions connected to Horadam sequence. That is, we define hyperbolic Horadam functions and present their hyperbolic and recursive properties. We give some geometrical properties of hyperbolic Horadam functions.
References
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- [2] Cieśliński, J.L., “New definitions of exponential, hyperbolic and trigonometric functions on time scales”, J. Math. Anal. Appl. 388, 8-22, (2012).
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- [6] Koçer, E.G., Tuğlu, N. and Stakhov, A., “Hyperbolic Functions with Second Order Recurrence Sequences”, Ars Combinatoria, 88, 65-81, (2008).
- [7] Lv, Y., Wang, G. and Chu, Y., “A note on Jordan type inequalities for hyperbolic functions”, Applied Mathematics Letters, 25, 505-508, (2012).
- [8] Pandir, Y. and Ulusoy, H., “New Generalized Hyperbolic Functions to Find New Exact Solutions of the Nonlinear Partial Differential Equations”, Journal of Mathematics Volume 2013, Article ID 201276, (2013).
- [9] De Spinadel, V.W., “From the Golden Mean to Chaos”, Nueva Libreria, 1998 (second edition, Nobuko, 2004).
- [10] Stakhov, A. P., “Gazale Formulas, a New Class of the Hyperbolic Fibonacci and Lucas Functions, and the Improved Method of the `Golden' Cryptography”, Academy of Trinitarism, No. 77-6567, 1-32, 2006.
- [11] Stakhov, A. P. and Tkachenko, I.S., “Hyperbolic Fibonacci trigonometry”, Rep Ukr. Acad. Sci. 7, 9-14 (1993).
- [12] Stakhov, A. P. and Rozin, B., “On a new class of hyperbolic functions”, Chaos, Solitons & Fractals, 23:(2), 379-389, (2005).
- [13] Stakhov, A. P. and Rozin, B., “The Golden Shofar”, Chaos, Solitons & Fractals, 26:(3), 677-684, (2005).
- [14] Stakhov, A. P. and Rozin, B., “The continuous functions for the Fibonacci and Lucas p-numbers”, Chaos, Solitons & Fractals, 28:(4), 1014-1025, (2006).
- [15] Stakhov, A. P. and Rozin, B., “The "golden" hyperbolic models of Universe”, Chaos, Solitons & Fractals, 34:(2), 159-171, (2007).
- [16] Taşçı, D. and Azman, H., “The k-Lucas Hyperbolic Functions, Communications in Mathematics and Applications”, 5:(1), 11-21, (2014).
- [17] Vajda, S., “Fibonacci & Lucas Numbers, and the Golden Section. Theory and Applications”, Ellis Horwood Limited. 1989.
- [18] Yang, C.-Y., “Inequalities on generalized trigonometric and hyperbolic functions”, J. Math. Anal. Appl. 419, 775-782, (2014).
Year 2019,
, 956 - 965, 01.09.2019
Mustafa Bahsı
,
Suleyman Solak
References
- [1] Baricza Á., Bhayo, B.A. and Pogány, T.K., “Functional inequalities for generalized inverse trigonometric and hyperbolic functions”, J. Math. Anal. Appl. 417, 244-259, (2014).
- [2] Cieśliński, J.L., “New definitions of exponential, hyperbolic and trigonometric functions on time scales”, J. Math. Anal. Appl. 388, 8-22, (2012).
- [3] Falcón, S. and Plaza, Á., “The k-Fibonacci hyperbolic functions”, Chaos, Solitons and Fractals, 38, 409-420, (2008).
- [4] Horadam, A.F., “Basic properties of a certain generalized sequence of numbers”, The Fibonacci Quart. 3, 161-176, (1965).
- [5] Klén, R., Vuorinen, M. and Zhang, X.H., “Inequalities for the generalized trigonometric and hyperbolic functions”, J. Math. Anal. Appl. 409, 521-529, (2014).
- [6] Koçer, E.G., Tuğlu, N. and Stakhov, A., “Hyperbolic Functions with Second Order Recurrence Sequences”, Ars Combinatoria, 88, 65-81, (2008).
- [7] Lv, Y., Wang, G. and Chu, Y., “A note on Jordan type inequalities for hyperbolic functions”, Applied Mathematics Letters, 25, 505-508, (2012).
- [8] Pandir, Y. and Ulusoy, H., “New Generalized Hyperbolic Functions to Find New Exact Solutions of the Nonlinear Partial Differential Equations”, Journal of Mathematics Volume 2013, Article ID 201276, (2013).
- [9] De Spinadel, V.W., “From the Golden Mean to Chaos”, Nueva Libreria, 1998 (second edition, Nobuko, 2004).
- [10] Stakhov, A. P., “Gazale Formulas, a New Class of the Hyperbolic Fibonacci and Lucas Functions, and the Improved Method of the `Golden' Cryptography”, Academy of Trinitarism, No. 77-6567, 1-32, 2006.
- [11] Stakhov, A. P. and Tkachenko, I.S., “Hyperbolic Fibonacci trigonometry”, Rep Ukr. Acad. Sci. 7, 9-14 (1993).
- [12] Stakhov, A. P. and Rozin, B., “On a new class of hyperbolic functions”, Chaos, Solitons & Fractals, 23:(2), 379-389, (2005).
- [13] Stakhov, A. P. and Rozin, B., “The Golden Shofar”, Chaos, Solitons & Fractals, 26:(3), 677-684, (2005).
- [14] Stakhov, A. P. and Rozin, B., “The continuous functions for the Fibonacci and Lucas p-numbers”, Chaos, Solitons & Fractals, 28:(4), 1014-1025, (2006).
- [15] Stakhov, A. P. and Rozin, B., “The "golden" hyperbolic models of Universe”, Chaos, Solitons & Fractals, 34:(2), 159-171, (2007).
- [16] Taşçı, D. and Azman, H., “The k-Lucas Hyperbolic Functions, Communications in Mathematics and Applications”, 5:(1), 11-21, (2014).
- [17] Vajda, S., “Fibonacci & Lucas Numbers, and the Golden Section. Theory and Applications”, Ellis Horwood Limited. 1989.
- [18] Yang, C.-Y., “Inequalities on generalized trigonometric and hyperbolic functions”, J. Math. Anal. Appl. 419, 775-782, (2014).