Zero and pole distribution of diagonal Padé approximants to the exponential function

The polynomials $P_n$ and $Q_m$ having degrees $n$ and $m$ respectively, with $P_n$ monic, that solve the approximation problem $$P_n(z)e^{-z+Q_m(z)={cal O left(z^{n+m+1right)$$ will be investigated for their asymptotic behaviour, 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 shall focus on the important diagonal case when $n=m$ and the polynomials $P_n$ and $Q_n$, as well as the remainder $E_{n,n(z)= P_n(z)e^{-z+Q_n(z)$ can be expressed in terms of Hankel and Bessel functions. The approximate location of the zeros of $P_n, Q_n$ and $E_{n,n$ are given in terms of the known zeros of certain Airy functions. An application is given in which the asymptotic information on the zeros is used to obtain an estimate in an approximation of the unit block function by means of the polynomials $P_n, Q_n$.