On Stakhov Functions and New Hyperboloid Surfaces

This paper presents an investigation into the generalization of hyperbolic Fibonacci sine and cosine functions, as well as Fibonacci spirals. Initially, we establish the main definitions and theoretically model them, listing several special cases. We then uncover fundamental results, including the De Moivre and Pythagorean formulas. Based on these new definitions, we introduce new classes of three-dimensional hyperboloid surfaces and compute their Gauss and mean curvatures. Notably, we demonstrate that these surfaces are geodesic.