International Journal of Mathematics and Mathematical Sciences
Volume 2004 (2004), Issue 58, Pages 3075-3101
doi:10.1155/S0161171204401215
A partial factorization of the powersum formula
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.
Abstract
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
ℜ of finite order such that the
αth power of each root of P is a solution of
ℜzα=0, and the coefficient functions of
ℜ all lie in the differential ring generated by the
coefficients of P and the integers ℤ. We call
ℜ 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.