Hyperbolic Functions with Second Order Recurrence Sequences

Abstract

In this paper, we introduce an extension of the hyperbolic Fibonacci and Lucas functions which were studied by Stakhov and Rozin. Namely, we define hyperbolic functions by second-order recurrence sequences and study their hyperbolic and recurrence properties. We give the corollaries for Fibonacci, Lucas, Pell, and Pell-Lucas numbers. We finalize with the introduction of some surfaces (the Metallic Shofars) that relate to the hyperbolic functions with the second-order recurrence sequences.