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.