© Hindawi Publishing Corp. POWERSUM FORMULA FOR DIFFERENTIAL RESOLVENTS

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 theα-resolvent. Finally, we use the powersum formula to rediscover Cockle’s differential resolvent of a cubic trinomial. 2000 Mathematics Subject Classification: 12H05, 13N15. 1. Introduction. It was proved in [4, Theorem 37, page 67] that for any integer q, a polynomial P(t) ≡ ∑Nk=0 (−1)N−keN−ktk of a single variable t whose coefficients {eN−k}Nk=0 lie in an ordinary differential ring R with derivation D possesses an ordinary linear differential α-resolvent and an ordinary linear differential q-resolvent, where α is a constant, transcendental over R. With no loss of generality, we assume that P i