Padé Approximants and Bounds to Series of Stieltjes

Abstract

Series of Stieltjes with nonzero radius of convergence R have been considered in this paper. It is well known that sequences of Padé approximants to these series may be defined which converge in the complex plane cut from −R to − ∞. It is shown that the Padé approximants satisfy inequalities between 0 and −R which are much more general than those already proved on the positive real axis. A new sequence of approximants is defined, which are closely related to the Padé approximants and which have very similar properties. The two sets of approximants may be used to determine the series of Stieltjes within certain limits for points on the real axis between 0 and −R, given only the first few coefficients of the power series expansion. The result is then extended to all points in the interior of a circle with center at the origin and radius R.