We consider a particular case of the general rational Hermite interpolation problem where the value of a function f is interpolated at some points, and where the formal series expansions at the origin of the rational interpolant and of f agree as far as possible. There are two different ways for constructing such a rational interpolant. Since the denominator of a Padétype approximant can be freely chosen, we will choose it in order that it also interpolates the function at some points. We thus obtain a Padétype rational interpolant (PTRI). Rational interpolation can be achieved by a barycentric formula where the weights can be freely chosen. We will choose them in order that this rational function matches the series f as far as possible. Thus, the Padétype barycentric interpolant (PTBI) constructed in this way possesses a Padétype property at the origin. A partial knowledge on the poles and/or the zeros of f can also be took into account. We can also consider the well known Padé approximants usually written under the form of a rational fraction or as the convergent of a certain continued fraction. But we will show that a Padé approximant can also be written under (at least) two different barycentric rational forms which depend on arbitrary parameters. Such an approximant will be called a barycentric Padé approximant (BPA). Then, we will write a Padé approximant under a partial fraction form, called a partial fraction Pad ́e approximant (PFPA), with the possibility to impose some poles and/or zeros. Numerical examples show the interest of these new representations, based in the computation of their coefficients by solving a linear system. This is a joint work with Claude Brezinski (Univ. of Lille, France).
