On polynomials related with Hermite - - Pade approximations to the exponential function

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) as x ! 0: We give a connection between the coeecients 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 n 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 innnity 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.