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Another way where they show up is in Lie theory. If you want to compute the derivative of the exponential map (of a Lie group) you encounter the function $x/(e^x - 1)$ quite inevitably. The already posted questions can partly be viewed as incarnations of this. THis results also in the appearence of the Bernoulli numbers in the BCH series, which is of course of fundamental importance far beyond the usage in Lie algebra theory...