Stieltjes analytic functions and higher order linear differential equations

TL;DR

This paper develops a comprehensive theory of Stieltjes-analytic functions, introducing Stieltjes monomials and polynomials, and applies it to solve higher order linear differential equations with constant coefficients.

Contribution

It introduces a new class of Stieltjes-analytic functions and provides methods to solve higher order Stieltjes differential equations, extending classical differential equation theory.

Findings

Defined Stieltjes monomials and polynomials.

Established properties of Stieltjes-analytic functions.

Solved higher order linear Stieltjes differential equations.

Abstract

In this work we develop a theory of Stieltjes-analytic functions. We first define the Stieltjes monomials and polynomials and we study them exhaustively. Then, we introduce the Stieltjes analytic functions locally, as an infinite series of these Stieltjes monomials and we study their properties in depth and how they relate to higher order Stieltjes differentiation. We define the exponential series and prove that it solves the first order linear problem. Finally, we apply the theory to solve higher order linear homogeneous Stieltjes differential equations with constant coefficients.