Padé-type approximants of entire functions and acceleration of convergence: Two examples

In this paper we deal with Padé-type approximants of entire functions. On two examples, we take the question: do these rational approximants converge more quickly than the partial sums to the entire functions? In particular, it is proved that the Padé-type approximants (k − 1,k) of exp(−z), whose generating polynomials are (z + 1k)k, ac accelerate the convergence of the partial sums σj=ok-1(1j!)(−z)j.