International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 58, Pages 3075-3101

A partial factorization of the powersum formula

John Michael Nahay

Swan Orchestral Systems, 25 Chestnut Hill Lane, Columbus 08022-1039, NJ, USA

Received 23 January 2004

Copyright © 2004 John Michael Nahay. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


For any univariate polynomial P whose coefficients lie in an ordinary differential field 𝔽 of characteristic zero, and for any constant indeterminate α, there exists a nonunique nonzero linear ordinary differential operator of finite order such that the αth power of each root of P is a solution of zα=0, and the coefficient functions of all lie in the differential ring generated by the coefficients of P and the integers . We call an α-resolvent of P. The author's powersum formula yields one particular α-resolvent. However, this formula yields extremely large polynomials in the coefficients of P and their derivatives. We will use the A-hypergeometric linear partial differential equations of Mayr and Gelfand to find a particular factor of some terms of this α-resolvent. We will then demonstrate this factorization on an α-resolvent for quadratic and cubic polynomials.