Lucas Polynomials and Power Sums

Abstract

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.