A partial factorization of the powersum formula (Q1777834)

Summary: For any univariate polynomial \(P\) whose coefficients lie in an ordinary differential field \(\mathbb{F}\) of characteristic zero, and for any constant indeterminate \(\alpha\), there exists a nonunique nonzero linear ordinary differential operator \(\mathfrak{R}\) of finite order such that the \(\alpha\)th power of each root of \(P\) is a solution of \(\mathfrak{R}z^\alpha=0\), and the coefficient functions of \(\mathfrak{R}\) all lie in the differential ring generated by the coefficients of \(P\) and the integers \(\mathbb{Z}\). We call \(\mathfrak{R}\) an \(\alpha\)-resolvent of \(P\). The author's powersum formula yields one particular \(\alpha\)-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 \(\alpha\)-resolvent. We will then demonstrate this factorization on an \(\alpha\)-resolvent for quadratic and cubic polynomials.