The polynomials $P_n$ and $Q_m$ having degrees $n$ and $m$ respectively, with $P_n$ monic, that solve the approximation problem $$ P_n(z)e^{-z+Q_m(z)={cal Oleft(z^{n+m+1right) $$ will be investigated for their asymptotic behavior, in particular in connection with the distribution of their zeros. The symbol ${cal O$ means that the left-hand side should vanish at the origin at least to the order $n+m+1$. This problem is discussed in great detail in a series of papers by Saff and Varga. In the present paper we show how their results can be obtained by using uniform expansions of integrals in which Airy functions are the main approximants. We give approximations of the zeros of $P_n$ and $Q_m$ in terms of zeros of certain Airy functions, as well of those of the remainder defined by $E_{n,m(z)= P_n(z)e^{-z+Q_m(z)$.

Padé approximation (msc 41A21), Asymptotic representations in the complex domain (msc 30E15), Bessel and Airy functions, cylinder functions, ${}_0F_1$ (msc 33C10), 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). Asymptotics and zero distribution of Padé polynomials associated with the exponential function. Modelling, Analysis and Simulation [MAS]. CWI.