On Polynomials Related with Hermite–Padé Approximations to the Exponential Function Original Research Article

We investigate the polynomialsPn,Qm, andRs, having degreesn, m, ands, respectively, withPnmonic, that solve the approximation problem [formula]We give a connection between the coefficients of each of the polynomialsPn,Qm, andRsand certain hypergeometric functions, which leads to a simple expression forQmin the casen=s. The approximate location of the zeros ofQm, whenn⪢mandnequals;s, are deduced from the zeros of the classical Hermite polynomial. Contour integral representations ofPn,Qm,Rs, andEnmsare given and, using saddle point methods, we derive the exact asymptotics ofPn,Qm, andRsasn,m, andstend 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 polynomialsPn,Qm, andRsfor the casen=s=40,m=45.