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.
