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.
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