Lucas polynomials and power sums

Abstract

The three - term recurrence xn + yn = (x + y) · (xn-1 + yn-1) - xy · (xn-2 + yn-2) allows to express xn + yn 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. © 2013 IEEE.