Hermite–Padé approximants of the Mittag-Leffler functions

The convergence rate of type II Hermite–Padé approximants for a system of degenerate hypergeometric functions $\{_1F_1(1,\gamma;\lambda_jz)\}_{j=1}^k$ is found in the case when the numbers $\{\lambda_j\}_{j=1}^k$ are the roots of the equation $\lambda^k=1$ or real numbers and $\gamma\in\mathbb C\setminus\{0,-1,-2,\dots\}$. More general statements are obtained for approximants of this type (including nondiagonal ones) in the case of $k=2$. The theorems proved in the paper complement and generalize the results obtained earlier by other authors.