Complex algebras on n-order polynomials and generalizations of trigonometry, oscillator model and Hamilton dynamics

A generator of the complex algebra within the framework of general formulation obeys the quadratic equation of the type e² = a₁e − a₀. In this paper we construct the general complex algebras of the n-th order where the generators obey n-order polynomial equation of the type eⁿ = a_{n - 1}e^{n - 1} − a_{n - 2}e^{n - 2} + ... + (−)^{n + 1}a₀, with real coefficients a_k, k = 0, 1, ... n − 1. This algebra induces a generalized trigonometry ((n + 1)-gonometry), subyacent to the n-th order oscillator model and to the n-th order Hamilton equations.