Denominators of coefficients of the Baker-Campbell-Hausdorff series

TL;DR

This paper derives an explicit formula for the common denominators of the rational coefficients in the Baker-Campbell-Hausdorff series, revealing their minimal size and connections to Bernoulli numbers, with implications for computational efficiency.

Contribution

It introduces a new explicit formula for calculating minimal common denominators of the series' coefficients, improving understanding and computation of the BCH series.

Findings

Derived an explicit formula for denominators up to degree 30

Found denominators are as small as possible, indicating optimality

Discovered connections with Bernoulli numbers and polynomials

Abstract

For the computation of terms of the Baker-Campbell-Hausdorff series $H = \log(e^Ae^B})$ some a priori knowledge about the denominators of the coefficients of the series can be beneficial. In this paper an explicit formula for the computation of common denominators for the rational coefficients of the homogeneous components of the series is derived. Explicit computations up to degree 30 show that the common denominators obtained by this formula are as small as possible, which suggests that the formula is in a sense optimal. The sequence of integers defined by the formula seems to be interesting also from a number-theoretic point of view. There is, e.g., a connection with the denominators of the Bernoulli numbers and the Bernoulli polynomials.