Asymptotics of Hermite–Padé degenerate hypergeometric functions

The asymptotic behavior of diagonal Hermite–Padé polynomials and diagonal Hermite–Padé approximations of type II for the system $\{_1F_1(1,\gamma;\lambda_jz)\}_{j=1}^k$, consisting of degenerate hypergeometric functions in which while the rest $\{\lambda_j\}_{j=1}^k$ are the roots of the equation $\lambda^k=1$, $\gamma$ — is a complex number belonging to the set $\mathbb{C}\setminus\{0,-1,-2,\dots\}$ was stated. The theorems complement known results of H. Padé, D. Braess, A.I. Aptekarev, H. Stahl, F. Wielonsky, W. Van Assche, A. B. J. Kuijlaars, A.P. Starovoitov, obtained for the case, where the $\{\lambda_p\}_{p=0}^k$ — different real numbers.