Asymptotics of Hermite–Padé approximation of exponential functions with complex multipliers in the exponent

Asymptotic properties of the diagonal Hermite–Padé approximants of type I for the system of exponentials $\{e^{\lambda_pz}\}^k_{p=0}$, in which $\lambda_0=0$, while the rest $\lambda_p$ are the roots of the equation $\xi^k=1$ are studied. The theorems complement known results of P. Borwein, F. Wielonsky, H. Stahl, A. V. Astafyeva, A. P. Starovoitov, obtained for the case, where the $\{\lambda_p\}^k_{p=0}$ — different real numbers.