Cauchy Functions for Dynamic Equations on a Measure Chain

We consider the nth-order linear dynamic equation Px(t) = ∑i = 0npi(t)x(σi(t)) = 0, where pi(t), 0 ≤ i ≤ n, are real-valued functions defined on T. We define the Cauchy function K(t, s) for this dynamic equation, and then we prove a variation of constants formula. One of our main concerns is to see how the Cauchy function for an equation is related to the Cauchy functions for the factored parts of the operator P. Finally we consider the equation Px(t) = ∑i = 0npix(σi(t)) = 0, where each of the pi's is a constant, and obtain a formula for the Cauchy function. For our main results we only consider the time scale T such that every point in T is isolated.