On the hyperbolic Horadam matrix functions

Year 2022, Volume: 51 Issue: 6, 1550 - 1562, 01.12.2022

Mustafa Bahşi , Efruz Özlem Mersin

https://doi.org/10.15672/hujms.1092305

Cited By: 2

Abstract

In this study, we introduce a new class of the hyperbolic matrix functions which are called symmetrical hyperbolic Horadam sine and cosine matrix functions and we present some hyperbolic and recursive properties of these new matrix functions. In addition, we introduce quasi-sine Horadam matrix function and also define the matrix form of the metallic shofars that related to the hyperbolic Horadam sine and hyperbolic Horadam cosine matrix functions.

Keywords

Horadam numbers, hyperbolic functions, hyperbolic matrix function

References

• [1] A.H. Al-Mohy and N.J. Higham, A New Scaling and Squaring Algorithm for the Matrix Exponential, SIAM J. Matrix Anal. Appl. 31 (3), 970-989, 2009.
• [2] M. Bahşi and S. Solak, On the Hyperbolic Fibonacci Matrix Functions, TWMS J. App. Eng. Math. 8 (2), 454-465, 2018.
• [3] M. Bahşi and S. Solak, Hyperbolic Horadam functions, Gazi Univ. J. Sci 32 (3), 956-965, 2019.
• [4] V.W. De Spinadel, From the Golden Mean to Chaos, Nueva Libreria, 1998, second edition, Nobuko, 2004.
• [5] E. Defez, J. Sastre, J.J. Ibanez, and P.A. Ruiz, Matrix Functions Solving Coupled Differential Models, Math. Comput. Model. 50 (5,6), 831-839, 2009.
• [6] E. Defez, J. Sastre, J.J. Ibanez, and P.A. Ruiz, Computing Matrix Functions Arising in Engineering Models with Orthogonal Matrix Polynomials, Math. Comput. Model. 57 (7,8), 1738-1743, 2013.
• [7] E. Defez, J. Sastre, J.J. Ibáñez and J. Peinado, Solving engineering models using hyperbolic matrix functions, Appl. Math. Model. 40 (4), 2837-2844, 2016.
• [8] G.I. Hargreaves, and N.J. Higham, Efficient Algorithms for the Matrix Cosine and Sine, Numer. Algorithms 40, 383-400, 2005.
• [9] N.J. Higham, Functions of Matrices: Theory and Computation, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2008.
• [10] N.J. Higham and M.I. Smith, Computing the Matrix Cosine, Numer. Algorithms 34, 13-16, 2003.
• [11] A.F. Horadam, Basic Properties of a Certain Generalized Sequence of Numbers, Fibonacci Q. 3, 161-176, 1965.
• [12] E.G. Koçer, N. Tuglu and A. Stakhov, Hyperbolic Functions with Second Order Recurrence Sequences, Ars Comb. 88, 65-81, 2008.
• [13] J. Sastre, J.J. Ibáñez, E. Defez, and P. Ruiz, Accurate matrix exponential computation to solve coupled differential models in engineering, Math. Comput. Model. 54, 1835- 1840, 2011.
• [14] J. Sastre, J.J. Ibáñez, P.A. Ruiz, and E. Defez, Efficient computation of the matrix cosine, Appl. Math. Comput. 219, 7575-7585, 2013.
• [15] A.P. Stakhov and B. Rozin, On a new class of hyperbolic functions, Chaos, Solitons & Fractals 23 (2), 379-389, 2005.
• [16] A.P. Stakhov and B. Rozin, The Golden Shofar, Chaos, Solitons & Fractals 26 (3), 677-684, 2005.
• [17] A.P. Stakhov and I.S. Tkachenko, Hyperbolic Fibonacci Trigonometry, Reports of the National Academy of Sciences of Ukraine, In Russian, 208 (7), 9-14, 1993.