This function comes in connection with exponentiation for matrix Lie groups, e.g. in computing the derivative of $\exp()$ away from the identity, or $\exp()$ of something in the affine group associated with a linear group. Does it have a name in the literature?
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I don't think it has a name. But if you are looking for something like a "tangent coset", then have a look at the Maurer-Cartan form. – geodude Apr 24 '14 at 21:30
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It is the exponential generating function of the harmonic numbers with -z instead of z. The Bernoulli numvers aruse naturally when inverting this around the origin. – Pedro Apr 24 '14 at 21:56