Apparent Singularities of D-finite Systems

TL;DR

This paper extends the concepts of singularities to D-finite systems, providing algorithms for detecting, removing apparent singularities, and computing formal solutions, thereby advancing the analysis of such systems.

Contribution

It introduces a generalized framework for singularities in D-finite systems and presents algorithms for their detection, removal, and solution computation.

Findings

Apparent singularities can be removed by adding solutions.

Algorithms for detecting and removing apparent singularities are developed.

Formal power series solutions at apparent singularities can be computed.

Abstract

We generalize the notions of singularities and ordinary points from linear ordinary differential equations to D-finite systems. Ordinary points of a D-finite system are characterized in terms of its formal power series solutions. We also show that apparent singularities can be removed like in the univariate case by adding suitable additional solutions to the system at hand. Several algorithms are presented for removing and detecting apparent singularities. In addition, an algorithm is given for computing formal power series solutions of a D-finite system at apparent singularities.