Asymptotics and zero distribution of Padé polynomials associated with the exponential function

The polynomials Pn and Qm having degrees n and m, respectively, with Pn monic, that solve the approximation problem Pn(z)e−z + Qm(z) = O(zn+m+1) will be investigated for their asymptotic behavior, in particular in connection with the distribution of their zeros. The symbol 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 Pn and Qm in terms of zeros of certain Airy functions, as well of those of the remainder defined by En,m(z) = Pn(z)e−z + Qm(z).