On Generalizing Cauchy's Theorem
by
M. I. Mostafa
Ain Shams University, Cairo

Using the formal derivative idea, we give a generalization for the Cauchy's Theorem relating to the factors of (x+y)^n-x^n-y^n. We determine the functions A(n), B(n) such that the polynomial A(n, a, b).(x+y)^n+B(n, a, b).(x^n+y^n) is expanded, for any natural number n, in terms of the the polynomials x+y and a.x^2+b.xy+a.y^2. As an application, we give an expansion for the polynomial (x+y)^n.(z^n+t^n)-(x^n+y^n).(z+t)^n in terms of the polynomials (x+y) and (xz-yt).(xt-yz). Also we give an interesting expansion for the polynomial x^n+y^n in terms of the polynomials x+y and x^2+3xy+y^2 that has the Lucas and Fibonacci sequences as the first and second coeffients respectively and the other terms are relating sequences.

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Date received: April 20, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacf-47.