Stirling numbers of the first kind are common in number theory and combinatorics; through Ewen’s sampling formula, these numbers enter into the calculation of several population genetics statistics, such as Fu’s Fs. In previous papers we have considered an asymptotic estimator for a finite sum of Stirling numbers, which enables rapid and accurate calculation of Fu’s Fs. These sums can also be viewed as cumulative distribution functions, leading directly to the possibility of an inversion function, where, given a value for Fu’s Fs, the goal is to solve for one of the input parameters. We solve this inversion using Newton iteration for small parameters. For large parameters, we have to extend our earlier obtained asymptotic results to solve the inversion problem asymptotically. Numerical experiments are given to show the efficiency of both solving the inversion problem and the expanded asymptotic estimator for sums of Stirling numbers.

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Mathematics of Computation
Centrum Wiskunde & Informatica, Amsterdam (CWI), The Netherlands

Chen, S.L, & Temme, N.M. (2022). A distribution function from population genetics statistics using Stirling numbers of the first kind: Asymptotics, inversion and numerical evaluation. Mathematics of Computation, 91(334), 871–885. doi:10.1090/mcom/3711