About the convergence rate Hermite - Pade approximants of exponential functions

This paper studies uniform convergence rate of Hermite - Pade approximants (simultaneous Pade approximants) {pi(j)(n,(m) over right arrow)(z)}(k)(j=1) for a system of exponential functions {e(lambda jz)}(k)(j=1), where {lambda(j)}(k)(j=1) are different nonzero complex numbers. In the general case a research of the asymptotic properties of Hermite - Pade 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 - Pade 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.