Lucas polynomials and power sums

The three - term recurrence x n + y n = (x + y) · (x n-1 + y n-1 ) - xy · (x n-2 + y n-2 ) allows to express x n + y n as a polynomial in the two variables x + y and xy. This polynomial is the bivariate Lucas polynomial. This identity is not as well known as it should be. It can be explained algebraically via the Girard - Waring formula, combinatorially via Lucas numbers and polynomials, and analytically as a special orthogonal polynomial. We shall briefly describe all these aspects and present an application from number theory.