Series extension: Predicting approximate series coefficients from a finite number of exact coefficients

TL;DR

This paper demonstrates that differential approximants can accurately predict future series coefficients from initial data, enhancing efficiency in combinatorics and statistical mechanics applications, even in complex asymptotic cases.

Contribution

It introduces the effective use of differential approximants for approximate series extension, improving predictions of subsequent coefficients from limited initial data.

Findings

Differential approximants predict next 100 coefficients from first 20 with useful accuracy.

Method saves computational time in enumeration problems and coefficient verification.

Effective even in cases with stretched exponential asymptotic behavior.

Abstract

Given the first 20-100 coefficients of a typical generating function of the type that arises in many problems of statistical mechanics or enumerative combinatorics, we show that the method of differential approximants performs surprisingly well in predicting (approximately) subsequent coefficients. These can then be used by the ratio method to obtain improved estimates of critical parameters. In favourable cases, given only the first 20 coefficients, the next 100 coefficients are predicted with useful accuracy. More surprisingly, this is also the case when the method of differential approximants does not do a useful job in estimating the critical parameters, such as those cases in which one has stretched exponential asymptotic behaviour. Nevertheless, the coefficients are predicted with surprising accuracy. As one consequence, significant computer time can be saved in enumeration problems where several runs would normally be made, modulo different primes, and the coefficients constructed from their values modulo different primes. Another is in the checking of newly calculated coefficients. We believe that this concept of approximate series extension opens up a whole new chapter in the method of series analysis.