A note on simultaneous polynomial approximation of exponential functions

Let α₁, …, α_m be distinct complex numbers and τ(1), …, τ(m) be non-negative integers. We obtain conditions under which the functions [formula omitted] form a perfect system, that is, for every set ρ(1), …, ρ(m) of non-negative integers, there are polynomials a₁ (z), …, a_m (z) with respective degrees exactly ρ(1)−1, …, ρ(m)−1, such that the function [formula omitted] has a zero of order at least ρ(1) + … + ρ(m)−1 at the origin. Moreover, subject to the evaluation of certain determinants, we give explicit formulae for the approximating polynomials a₁ (z), …, a_m (z).