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

Abstract.

A generator of the complex algebra within the framework of general formulation obeys the quadratic equation of the type e ² = a ₁ e − a ₀. 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 e ⁿ = a _n - 1 e ^n - 1 − a _n - 2 e ^n - 2 + ... + (−)^n + 1 a ₀, 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.