Interlacing, integrals of Hurwitz series and differential equations

TL;DR

This paper explores the algebraic structures of Hurwitz series, introducing unlacing and integral operations, and applies these to analyze the exponential function and reduction of order in differential equations.

Contribution

It introduces the concept of unlacing as an inverse to interlacing of Hurwitz series and develops their properties, providing new tools for manipulating Hurwitz series.

Findings

Unlacing of Hurwitz series is defined and its properties are established.

Explicit formulas for multiplication involving interlacing and integrals are derived.

The exponential function is analyzed within Hurwitz series, enabling reduction of order in differential equations.

Abstract

In this paper, we first introduce the unlacing of Hurwitz series, which can be viewed as an inverse of interlacing, and develop the basic properties of unlacing, interlacing and integral of Hurwitz series. We then show that the multiplication of interlacing of Hurwitz series by basis elements can be also rewritten as an interlacing, and provide the explicit multiplication formula in terms of integrals and Hadamard products. Finally, we investigate the natural exponential function $\exp$, and realize the method of reduction of order in the ring of Hurwitz series by using $\exp$.