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

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.