Interlacing, integrals of Hurwitz series and differential equations
Abstract
The ring of Hurwitz series introduced by Keigher has many important applications in differential algebras and differential equations. In particular, solutions of linear homogeneous differential equations can be expressed by an interlacing of Hurwitz series.
This paper introduces the notion of unlacing of Hurwitz series, as an inverse of interlacing. Some basic properties of unlacing, interlacing and integral of Hurwitz series are discussed. We then show that an interlacing of Hurwitz series multiplied by arbitrary elements can be rewritten as a new interlacing of Hurwitz series, and thus give a positive answer to the open problem proposed by Gao and Keigher (2017). Finally, we use the exponential function to realize the method of reduction of order in the ring of Hurwitz series.
