For representing a function f by rational functions,the information on the function f can consist either of the first coefficients of its Taylor series expansion around zero,or in its values at some points of the complex plane.In the first case,Padé approximants can be used [1].They are rational functions whose series expansion around zero coincides with the series f as far as possible.Both the denominator and the numerator are fully determined by this condition.In Padé–type approximation [2],the denominator can be arbitrarily chosen and,then,the coefficients of the numerator are obtained by imposing the preceding approximation conditions.In the second case,an interpolating rational function can be built using Thieles formula.It achieves the maximum number of interpolation conditions,but no choice is left in its construction.On the other hand,when the degrees of the denominator and of the numerator are the same,writing the rational interpolant under a barycentric form allows to freely choose the weights appearing in this formula.In the first part of this talk,starting either from Padé–type approximants or from barycentric rational interpolants,we will show how to construct rational functions possessing both properties,that is interpolating f at some points of the complex plane,and whose series expansion around zero coincides with the Taylor series f as far as possible.Numerical examples will be given [3].In the second part of the talk,we will show how to write Padé approximants 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 [4].