A partial factorization of the powersum formula

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 α th power of each root of P is a solution of ℜ z α = 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.