Superexponentials: A Generalization of Hyperbolic and Trigonometric Functions

We construct and explore the properties of a generalization of hy- perbolic and trigonometric functions we cal l superexponentials. The general ization is based on the characteristic second-order differential equations (DE) these functions satisfy, and leads to functions satisfying analogous mth 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 superexponentials.