Journal article
Analysis of series expansions for non-algebraic singularities
AJ Guttmann
Journal of Physics A Mathematical and Theoretical | Published : 2015
Abstract
An underlying assumption of existing methods of series analysis is that singularities are algebraic. Functions with such singularities have their nth coefficient behaving asymptotically as A · μn · ng Recently, a number of problems in statistical mechanics and combinatorics has been encountered in which the coefficients behave asymptotically as B · μn μ1 nσ · ngwhere typically σ is a simple, rational number between 0 and 1. Identifying this behaviour, and then extracting estimates for the critical parameters B, μ, μ1, σ presents a significant numerical challenge. We describe methods developed to meet this challenge.
Grants
Awarded by Australian Research Council
Funding Acknowledgements
I would like to thank Nick Beaton, Andrew Conway, Iwan Jensen, Einar Steingrimsson and Stu Whittington for careful reading of the manuscript, which resulted in very substantial improvement. I have benefited from discussions with Mireille Bousquet-Melou and Bruno Salvy on questions of singularities and asymptotic behaviour of coefficients, for which I am grateful. I'm also grateful to Robin Pemantle and Brendan McKay who sorted out the asymptotic behaviour of coefficients in the generating function for compressed Dyck paths, as discussed in section 7. I would also like to thank the Australian Research Council who have supported this work through grant DP120100939.