Padé approximants of the Mittag-Leffler functions

It is shown that for $m\le n$ the Padé approximants $\{\pi_{n,m}(\,\cdot\,;F_{\gamma})\}$, which locally deliver the best rational approximations to the Mittag-Leffler functions $F_\gamma$, approximate the $F_\gamma$ as $n\to\infty$ uniformly on the compact set $D=\{z:|z|\le1\}$ at a rate asymptotically equal to the best possible one. In particular, analogues of the well-known results of Braess and Trefethen relating to the approximation of $\exp{z}$ are proved for the Mittag-Leffler functions.
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