We investigate the polynomials $P_n,,Q_m$ and $R_s$, having degrees $n,,m$ and $s$ respectively, with $P_n$ monic, that solve the approximation problem $$E_{nms(x):=P_n(x)e^{-2x+Q_m(x)e^{-x+R_s(x)=O(x^{n+m+s+2) {quad rm as quad x rightarrow 0. $$ We give a connection between the coefficients of each of the polynomials $P_n,,Q_m$ and $R_s$ and certain hypergeometric functions, which leads to a simple expression for $Q_m$ in the case $n=s$. The approximate location of the zeros of $Q_m$, when $ngg m$ and $n=s$, are deduced from the zeros of the classical Hermite polynomial. Contour integral representations of $P_n,,Q_m,,R_s$ and $E_{nms$ are given and, using saddle point methods, we derive the exact asymptotics of $P_n,,Q_m$ and $R_s$ as $n,,m$ and $s$ tend to infinity through certain ray sequences. We also discuss aspects of the more complicated uniform asymptotic methods for obtaining insight into the zero distribution of the polynomials, and we give an example showing the zeros of the polynomials $P_n,,Q_m$ and $R_s$ for the case $n=s=40, m=45$.

Padé approximation (msc 41A21), Asymptotic representations in the complex domain (msc 30E15), Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) (msc 30C15), Asymptotic approximations, asymptotic expansions (steepest descent, etc.) (msc 41A60)
Modelling, Analysis and Simulation [MAS]
Computational Dynamics

Driver, K.A, & Temme, N.M. (1997). On polynomials related with Hermite-Padé approximations to the exponential function. Modelling, Analysis and Simulation [MAS]. CWI.