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 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 Ahypergeometric 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.