Asymptotic behaviour of the Hadamard determinants and the behaviour of the rows of the Padé and Chebyshev tables for a sum of exponentials

For function $f(z)=\sum _{j=1}^k e^{\lambda _j z}$ asymptotic equalities for the Hadamard determinants constructed from its Taylor coefficients are established. Using them, the asymptotics of the deviations from $f(z)$ of its Padé approximations $\Pi _{n,m}(z)$ and of the corresponding rational functions of best uniform approximation $r_{n,m}^*(z)=p_n^*(z)=q_m^*(z)$ is found for $m$ fixed as $n \to \infty$.