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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

We will determine the number of powers of α that appear with nonzero coefficient in an α-power linear differential resolvent of smallest possible order of a univariate polynomial P(t) whose coefficients lie in an ordinary differential field and whose distinct roots are differentially independent over constants. We will then give an upper bound on the weight of an α-resolvent of smallest possible weight. We will then compute the indicial equation, apparent singularities, and Wronskian of the Cockle α-resolvent of a trinomial and finish with a related determinantal formula.2000 Mathematics Subject Classification: 12H05, 13N15.

We will prove that we can specialize the indeterminate α in a linear differential α-resolvent of a univariate polynomial over a differential field of characteristic zero to an integer q to obtain a q-resolvent. We use this idea to obtain a formula, known as the powersum formula, for the terms of the α-resolvent. Finally, we use the powersum formula to rediscover Cockle's differential resolvent of a cubic trinomial.2000 Mathematics Subject Classification: 12H05, 13N15.

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