Hermite–Pade approximants of the system Mittag-Leffler functions

The paper deals with asymptotic properties of Hermite integrals. In particular, the asymptotics of diagonal Hermite–Pade approximations $\pi^j_{kn,kn}(z;e^{j\xi})$ for the system of exponents $\{e^{jz}\}_{j=1}^k$ are determined when $j=1,2,\dots,k$ and $n\to\infty$. Similar results are proved for the system of confluent hypergeometric functions $\{_1F_1(1;\gamma;jz)\}_{j=1}^k$.