New representations of Pad\'e and Pad\'e-type approximants

TL;DR

This paper introduces new barycentric and partial fraction representations for Padé and Padé-type approximants, enhancing their flexibility and potential applications in rational approximation.

Contribution

It presents alternative barycentric and partial fraction forms for Padé approximants, expanding their representation options beyond the traditional rational form.

Findings

New barycentric forms for Padé approximants

Partial fraction representations depending on free parameters

Enhanced flexibility in rational approximation methods

Abstract

Pad\'e approximants are rational functions whose series expansion match a given series as far as possible. These approximants are usually written under a rational form. In this paper, we will show how to write them also under two different barycentric forms, and under a partial fraction form, depending on free parameters. According to the choice of these parameters, Pad\'e-type approximants can be obtained under a barycentric or a partial fraction form.