Superexponentials: A Generalization of Hyperbolic and Trigonometric Functions

We construct and explore the properties of a generalization of hyperbolic and trigonometric functions we call superexponentials. The generalization is based on the characteristic second-order differential equations (DE) these functions satisfy, and leads to functions satisfying analogous math order equations and having many properties analogous to the usual hyperbolic and trigonometric functions. Roots of unity play a key role in providing the periodicity resulting in various properties. We also show how these functions solve the general initial value problem for the differential equations y (n) = y, and a look at the power series expansions reveal surprisingly simple patterns that clarify the properties of the super exponentials.