New representations of Padé, Padé-type, and partial Padé approximants

Padé approximants are rational functions, with a denominator which does not vanish at zero, and 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, Padé-type approximants can be obtained under a barycentric or a partial fraction form.