The Matrix Form of the k-Pell Hyperbolic Functions

In the present paper, we introduce the matrix form of the k-Pell hyperbolic sine and cosine functions, along with their symmetrical forms. We examine their recurrence and hyperbolic properties, including Pythagorean, de Moivre, Catalan, Cassini, and d’Ocagne identities, as well as various sum and difference identities. Additionally, we define the quasi-sine k-Pell matrix functions and the matrix form of the three-dimensional k-Pell spiral, and investigate some of their fundamental properties. Matrix functions play a significant role in various scientific fields, particularly in mathematics and engineering, as they frequently arise in the solutions of differential equations. The findings presented contribute to the development of special matrix functions and their potential applications.