Barycentric Pad\'e approximation

TL;DR

This paper demonstrates how any Padé approximant of a formal power series can be expressed in two barycentric rational forms with flexible parameters, enhancing understanding of rational approximation representations.

Contribution

It introduces two barycentric rational forms for Padé approximants, allowing flexible parameter choices and providing new insights into their structure.

Findings

Any Padé approximant can be written in two barycentric forms.

The barycentric forms depend on p+q+1 parameters that can be chosen arbitrarily.

This representation broadens the understanding of rational approximation methods.

Abstract

In this paper, we show how any Pad\'e approximant $[p/q]_f$ of a formal power series $f$ can be written under two different barycentric rational forms. These form depend on $p+q+1$ parameters which can be almost arbitrarily chosen.