Interlacing of Hurwitz series

In this paper, we introduce the notion of interlacing of Hurwitz series. We begin by reviewing some important properties of the ring of Hurwitz series over a commutative ring A of arbitrary characteristic, and we introduce and investigate properties of the maps exp and log. We show that solutions of linear homogeneous differential equations with constant coefficients from the ring A can be described simply as interlacings of solutions of a first order system of differential equations. We give several examples to illustrate this result, and we conclude by defining and investigating properties of trigonometric functions using interlacings of Hurwitz series.