A New Approach to Polynomial Identities

Abstract

Using the formal derivative idea, we give a generalization for the Cauchy’s Theorem relating to the factors of (x + y)n−xn− yn. We determine the polynomials A(n, a, b) and B(n, a, b) such that the polynomial $$ A(n, a, b)(x + y)^n + B(n, a, b)(x^n + y^n) $$ can be expanded, for any natural number n, in terms of the polynomials x+y and ax2+bxy + ay2. We show that the coefficients of this expansion are intimately related to the Fibonacci, Lucas, Mersenne and Fermat sequences. As an application, we give an expansion for $$ (x^n + y^n){(z + t)^n}-{(x + y)^n}(z^n + t^n) $$ as a polynomial in x+y and (xz −yt)(xt−yz). We use this expansion to find closely related identities to the sums of like powers. Also, we give two interesting expansions for the polynomials \(\frac{x^{n}-y^{n}}{x-y}\) and xn+yn that we call Fibonacci expansions and Lucas expansions respectively. We prove that the first coefficient of these two expansions is a Fibonacci sequence and a Lucas sequence respectively and the other coefficients are related sequences. Finally we give a generalization for all the previous results.