Smallest common denominators for the homogeneous components of the Baker-Campbell-Hausdorff series

TL;DR

This paper proves that a previously derived formula for common denominators of the BCH series components is actually minimal and introduces an efficient integer-based algorithm for computing BCH coefficients.

Contribution

It establishes the minimality of the common denominators formula and presents a new integer arithmetic algorithm for BCH coefficient computation.

Findings

The formula for common denominators is proven to be the smallest possible.

An efficient integer-based algorithm for BCH coefficients is introduced.

The algorithm improves computational efficiency by avoiding rational arithmetic.

Abstract

In a recent paper the author derived a formula for calculating common denominators for the homogeneous components of the Baker-Campbell-Hausdorff (BCH) series. In the present work it is proved that this formula actually yields the smallest such common denominators. In an appendix a new efficient algorithm for computing coefficients of the BCH series is presented, which is based on these common denominators, and requires only integer arithmetic rather than less efficient rational arithmetic.