The 𝐴𝑡𝑒𝑏(ℎ)-functions and their properties

The Ateb(h) -functions are inversions of incomplete Beta-functions. They are the solutions of normal mode vibrations of certain nonlinear multi-degree-of-freedom systems just as the trigonometric functions yield the normal mode vibrations of the corresponding linear systems. Like elliptic functions, the Ateb(h) -functions depend on a parameter. The Ateb -functions reduce to trigonometric functions, and the Atebh -functions to hyperbolic functions when the parameter is 1. When the parameter is 2, the Ateb -functions become elliptic functions. A number of properties of the Ateb(h) -functions, such as identities, derivatives, integrals, differential equations satisfied by them, etc., are given.