On the Bihyperbolic \(k\)-Fibonacci Numbers

In this paper, we extend the concept of bihyperbolic numbers to the \(k\)-Fibonacci numbers. We will find the main formulas that affect these numbers, such as the recurrence formulas and the sum of the first \(n\) terms, as well as the identities of Binet, Catalan and d’Ocagne. We finish the study by finding the generating function of the bihyperbolic \(k\)-Fibonacci numbers.