On the Generalized \(k\)-Fibonacci Hyperbolic Functions

Abstract

The hyperbolic Fibonacci function, which is the continuous extension of Binet’s formula for the Fibonacci number, transforms the Fibonacci number theory into a “continuous” theory because every identity for the hyperbolic Fibonacci function has its discrete analogy in the framework of the Fibonacci number. In this new paper, we define three important generalizations of the \(k\)-Fibonacci sine, cosine, and quasi-sine hyperbolic functions and then carry over many concepts and techniques that we learned in a standard setting for the \(k\)-Fibonacci sine, cosine, and quasi-sine hyperbolic functions to the generalizations of these functions.