Analysis of series expansions for non-algebraic singularities

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.