About the convergence rate Hermite – Pade approximants of exponential functions

This paper studies uniform convergence rate of Hermite\,--\,Pad\'eapproximants (simultaneous Pad\'e 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 nonzerocomplex numbers. In the general case a research of theasymptotic properties of Hermite\,--\,Pad\'e approximants is arather complicated problem. This is due to the fact that in theirstudy mainly asymptotic methods are used, in particular,the saddle-point method. An important phase in the applicationof this method is to find a special saddle contour (the Cauchyintegral theorem allows to choose an integration contour ratherarbitrarily), according to which integration should be carriedout. 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\,--\,Pad\'eapproximants, that is based on the Taylor theorem and heuristicconsiderations underlying the Laplace and saddle-point methods,as well as on the multidimensional analogue of the Van Rossumidentity that we obtained. The proved theorems complement andgeneralize the known results by other authors.