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 $A\cdot \hat{I}{¼}^n\cdot n^g.$ Recently, a number of problems in statistical mechanics and combinatorics have been encountered in which the coefficients behave asymptotically as $B\cdot \hat{I}{¼}^n\cdot \hat{I}{¼}_1^{n^{\ddot{I}}ƒ}\cdot n^g,$ where typically $\ddot{I}ƒ=\frac{1}{2}$ or $\frac{1}{3}.$ Identifying this behaviour, and then extracting estimates for the critical parameters $B,\hat{I}¼,\hat{I}{¼}_1,\ddot{I}ƒ,\mathrm{and}g$ presents a significant numerical challenge. We describe methods developed to meet this challenge.

35 pages, 11 figures (3 small typos corrected)

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