On recursive hyperbolic functions in Fibonacci-Lucas sense

Abstract

The hyperbolic Fibonacci and hyperbolic Lucas functions have been introduced before and have been improved to functions of the symmetrical form. In this paper, we generalize the mentioned definitions, which will be called Horadam hyperbolic sine function $ \left( HSF \right) $ and Horadam hyperbolic cosine function $ \left( HCF \right) $. Further, we present many identities and hyperbolic properties of our new definitions.

Keywords

• Horadam hyperbolic function
• Horadam spiral
• quasi-sine Horadam function
• Honsberger formula
• Vajda’s identity

References

1. [1] H. Bulut, Y. Pandir, and H.M. Baskonus, Symmetrical hyperbolic Fibonacci function solutions of generalized Fisher equation with fractional order, AIP Conf. Proc. 1558 (1), 1914–1918, 2013.
2. [2] S.M. Ege and E. Misirli, The modified Kudryashov method for solving some fractional-order nonlinear equations, Adv. Differ. Equ. 2014, Art. No. 135, 2014.
3. [3] S. Falcón and Á. Plaza, On the Fibonacci k-numbers, Chaos Solitons Fractals, 32 (5), 1615–1624, 2007.
4. [4] S. Falcón and Á. Plaza, The k-Fibonacci hyperbolic functions, Chaos Solitons Fractals, 38 (2), 409–420, 2008.
5. [5] A.F. Horadam, A generalized Fibonacci sequence, Amer. Math. Monthly, 68 (5), 455– 459, 1961.
6. [6] A.F. Horadam, Generating functions for powers of a certain generalized sequence of numbers, Duke. Math. J. 32 (3), 437–446, 1965.
7. [7] A.F. Horadam, Basic properties of a certain generalized sequence of numbers, Fibonacci Quart. 3 (3), 161–176, 1965.
8. [8] A.F. Horadam, Special properties of the sequence ${W_n}\left( {a,b;p,q} \right)$, Fibonacci Quart. 5 (3), 424–434, 1967.

9. [9] T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley and Sons, Inc., Hoboken, NJ, 2019.
10. [10] Y. Pandir, Y. Gurefe, and E. Misirli, A new approach to Kudryashov’s method for solving some nonlinear physical models, Int. J. Phys. Sci. 7 (21), 2860–2866, 2012.
11. [11] S.S. Ray, New analytical exact solutions of time fractional KdV-KZK equation by Kudryashov methods, Chinese Physics B. 25 (4), 040204, 2016.
12. [12] A.P. Stakhov, Hyperbolic Fibonacci and Lucas functions: a new mathematics for the living nature, ITI, Vinnitsa, 2003.
13. [13] A.P. Stakhov and S. Aranson, Hyperbolic Fibonacci and Lucas functions, “Golden” Fibonacci goniometry, Bodnar’s geometry, and Hilbert’s fourth problem, Appl. Math. 2, 74–84, 2011.
14. [14] A.P. Stakhov and B. Rozin, On a new class of hyperbolic functions, Chaos Solitons Fractals, 23, 379–389, 2005.
15. [15] A.P. Stakhov and B. Rozin, The Golden Shofar, Chaos Solitons Fractals, 26 (3), 677-684, 2005.
16. [16] A.P. Stakhov and B. Rozin, The “golden” hyperbolic models of Universe, Chaos Solitons Fractals, 34 (2), 159–171, 2007.
17. [17] A.P. Stakhov and I.S. Tkachenko, Hyperbolic Fibonacci trigonometry, Rep. Ukr. Acad. Sci. 208, 9–14, 1993.