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 pr oblem E-nms(x) := P-n(x) e(-2x) + Q(m)(x) e(-x) + R-s(x) = C(x(n+m+s+2 )) as x --> 0. We give a connection between the coefficients of each o f the polynomials P-n, Q(m), and R-s and certain hypergeometric functi ons, which leads to a simple expression for Q(m) in the case n = s. Th e approximate location of the zeros of Q(m), when n much greater than m and n = s, are deduced from the zeros of the classical Hermite polyn omial. Contour integral representations of P-n, Q(m), R-s, and E-nms a re given and, using saddle point methods, we derive the exact asymptot ics of P-n, Q(m), and R-s as n, m, and s tend to infinity through cert ain ray sequences. We also discuss aspects of the more complicated uni form asymptotic methods for obtaining insight into the zero distributi on 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. (C) 1 998 Academic Press.