About the convergence rate Hermite – Padé approximants of exponential functions

This paper studies uniform convergence rate of Hermite – Padé approximants (simultaneous Padé approximants) $\{\pi^j_{n,\overrightarrow{m}}(z)\}_{j=1}^k$ for a system of exponential functions $\{e^{\lambda_jz}\}_{j=1}^k$, where $\{\lambda_j\}_{j=1}^k$ are different nonzero complex numbers. In the general case a research of the asymptotic properties of Hermite – Padé approximants is a rather complicated problem. This is due to the fact that in their study mainly asymptotic methods are used, in particular, the saddle-point method. An important phase in the application of this method is to find a special saddle contour (the Cauchy integral theorem allows to choose an integration contour rather arbitrarily), according to which integration should be carried out. Moreover, as a rule, one has to repy only on intuition. In this paper, we propose a new method to studying the asymptotic properties of Hermite – Padé approximants, that is based on the Taylor theorem and heuristic considerations underlying the Laplace and saddle-point methods, as well as on the multidimensional analogue of the Van Rossum identity that we obtained. The proved theorems complement and generalize the known results by other authors.