MULTIVARIATE NEWTON SUMS: IDENTITIES AND GENERATING FUNCTIONS

ABSTRACT

This paper is devoted to present, first, a family of formulas extending to the multivariate case the classical Newton (or Newton–Girard) Identities relating the coefficients of a univariate polynomial equation with its roots through the Newton Sums and, secondly, the Generating Functions associated to the new introduced Newton Sums of the multivariate case. As a by-product the kinds of systems accepting these Newton Identities are also characterized together with those allowing the Newton Sums to be computed in an inductive way directly from the coefficients of the polynomial system under consideration.

ACKNOWLEDGMENTS

We thank E. Becker for his comments and support for this work since he proposed to us the question about the existence of generating functions for the Multivariate Newton Sums and other kinds of similar multivariate symmetric functions. Partially supported by the grant DGESIC PB 98-0713-C02-02 (Ministerio de Educacio´n y Cultura).