AI 中文总结
本文提出了Baker-Campbell-Hausdorff公式的Eichler证明的变体，通过四指数乘积结合性简化证明，得到初等简洁的短证明，还推出不含显式伯努利数的递推格式。
AI 中文摘要
Eichler基于三个指数乘积的结合性，给出了Baker-Campbell-Hausdorff公式的一个初等但复杂的证明。本文表明，通过考虑四个指数乘积的结合性，可避开Eichler证明的复杂之处，得到一个更短、同时兼具初等性与简洁性的证明。作为该证明的推论，还推导出了Baker-Campbell-Hausdorff公式的一个递推格式，其中显著不含显式伯努利数。
英文摘要
Eichler gave an elementary but involved proof of the Baker-Campbell-Hausdorff formula, based on the associativity of the product of three exponentials. We show that by considering the associativity of the product of four exponentials, one can bypass the complications of Eichler's proof to arrive at a shorter proof that is both elementary and straightforward. As a corollary of the proof, a recursive scheme for the Baker-Campbell-Hausdorff formula is derived, in which explicit Bernoulli numbers are notably absent.
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