Analysis of series expansions for non-algebraic singularities

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 $A \cdot μ^n \cdot n^g.$ Recently, a number of problems in statistical mechanics and combinatorics have been encountered in which the coefficients behave asymptotically as $B \cdot μ^n \cdot μ_1^{n^σ} \cdot n^g,$ where typically $σ= \frac{1}{2}$ or $\frac{1}{3}.$ Identifying this behaviour, and then extracting estimates for the critical parameters $B, \,\, μ, \,\, μ_1, \,\, σ, \,\, {\rm and} \,\, g$ presents a significant numerical challenge. We describe methods developed to meet this challenge.

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