Finding Hyperexponential Solutions of Linear ODEs by Numerical Evaluation

TL;DR

This paper introduces a novel numerical algorithm for finding hyperexponential solutions of linear differential equations with polynomial coefficients by evaluating formal series solutions at a common point.

Contribution

The paper presents a new numerical method that interprets formal series solutions as analytic functions and uses their evaluations to identify hyperexponential solutions.

Findings

Effective algorithm for hyperexponential solutions

Numerical evaluation at common points simplifies solution detection

Applicable to linear ODEs with polynomial coefficients

Abstract

We present a new algorithm for computing hyperexponential solutions of ordinary linear differential equations with polynomial coefficients. The algorithm relies on interpreting formal series solutions at the singular points as analytic functions and evaluating them numerically at some common ordinary point. The numerical data is used to determine a small number of combinations of the formal series that may give rise to hyperexponential solutions.