Modified Padé–Borel Summation

Abstract

We revisit the problem of calculating amplitude at infinity for the class of functions with power-law behavior at infinity by means of a resummation procedure based on the truncated series for small variables. Iterative Borel summation is applied by employing Padé approximants of the “odd” and “even” types modified to satisfy the power-law. The odd approximations are conventional and are asymptotically equivalent with an odd number of terms in the truncated series. Even approximants are new, and they are constructed based on the idea of corrected approximants. They are asymptotically equivalent to the even number of terms in truncated series. Odd- and even-modified Padé approximants could be applied with and without a Borel transformation. The four methods are applied to some basic examples from condensed matter physics. We found that modified Padé–Borel summation works well in the case of zero-dimensional field theory with fast-growing coefficients and for similar examples. Remarkably, the methodology of modified Padé–Borel summation appears to be extendible to the instances with slow decay or non-monotonous behavior. In such situations, exemplified by the problem of Bose condensation temperature shift, the results are still very good.