Hermitian Approximation of Two Exponents

We study the asymptotic properties of Hermite–Pade approximants $\{\pi_{n,\,m}^j(z;\,e^{\lambda_j\,\xi})\}_{j=1}^2$ for a system consisting of functions $\{e^{\lambda_1 z},e^{\lambda_2 z}\}$. In particular, we determine asymptotic behavior of differences $e^{\lambda_j\,z}-\pi_{n,\,m}^j(z;\,e^{\lambda_j\,\xi})$ for $j=1,2$ and $n\rightarrow\infty$ for any complex number $z$. The obtained results supplement research of Pade, Perron, D. Braess and A. I. Aptekarev dealing with study of the convergence of joinnt Pade approximants for systems of exponents.