Analysis of series expansions for non-algebraic singularities

TL;DR

This paper develops methods to analyze series coefficients with non-algebraic singularities, common in statistical mechanics and combinatorics, where coefficients exhibit complex asymptotic behavior involving fractional powers.

Contribution

It introduces novel techniques for identifying and estimating parameters in series with non-algebraic singularities, extending beyond traditional algebraic analysis methods.

Findings

Methods successfully identify asymptotic parameters in complex series.

Enhanced accuracy in estimating critical parameters like , , , , and g.

Applicable to problems in statistical mechanics and combinatorics.

Abstract

Existing methods of series analysis are largely designed to analyse the structure of algebraic singularities. Functions with such singularities have their $n^{th}$ coefficient behaving asymptotically as $A \cdot \mu^n \cdot n^g.$ Recently, a number of problems in statistical mechanics and combinatorics have been encountered in which the coefficients behave asymptotically as $B \cdot \mu^n \cdot \mu_1^{n^\sigma} \cdot n^g,$ where typically $\sigma = \frac{1}{2}$ or $\frac{1}{3}.$ Identifying this behaviour, and then extracting estimates for the critical parameters $B, \,\, \mu, \,\, \mu_1, \,\, \sigma, \,\, {\rm and} \,\, g$ presents a significant numerical challenge. We describe methods developed to meet this challenge.