A new method to sum divergent power series: educated match

Abstract

We present a method to sum Borel- and Gevrey-summable asymptotic series by matching the series to be summed with a linear combination of asymptotic series of known functions that themselves are scaled versions of a single, appropriate, but otherwise unrestricted, function $\Phi$. Both the scaling and linear coefficients are calculated from Pad\'e approximants of a series transformed from the original series by $\Phi$. We discuss in particular the case that $\Phi$ is (essentially) a confluent hypergeometric function, which includes as special cases the standard Borel-Pad\'e and Borel-Leroy-Pad\'e methods. A particular advantage is the mechanism to build knowledge about the summed function into the approximants, extending their accuracy and range even when only a few coefficients are available. Several examples from field theory and Rayleigh-Schr\"odinger perturbation theory illustrate the method.