The exponential generating functions for the Stirling numbers 2nd kind are the n'th powers of $f(x)=\exp(x)-1$ (where this is understood as formal power series, Abramowitz&Stegun, 26.8.12).
Generalizations of the Stirling numbers 2nd kind have sometimes been discussed ; if we take negative powers $n\lt 0$ we get powers of $ {1 \over \exp(x)-1 } $ as exponential generating functions the resulting coefficients are then compositions of bernoulli-numbers (even better: of zeta-values at negative arguments) ; but the logical recomputation for the Stirlingnumbers 2nd kind involve then formally divisions by $(-1)!$ and so that kind of generalizations from the left bottom area in the matrix below come out to be infinitesimals, and any manipulation with this should be complicated/nontrivial.
Q: Does someone know, whether (and then where) this type of generalizations has been discussed?
The matrix below (truncated, should be of infinite size) shows the extension of the scaled Stirlingnumbers 2nd kind to negative column- and rowindices; the set of coefficients along the column c has the generating function $(\exp(x)-1)^c$.
Here the unscaled coefficients; in the bottom right-section we see the Stirling numbers 2nd kind, in the bottom left area the "generalized" with negative column-index; the symbol "z" indicates the division by $(-1)!$ and we have thus infinitesimal expressions. Interestingly, in the top-left segment we find the Stirling numbers 1st kind (factorially rescaled).
Note, this is a specific detail in the same area as in my earlier question here
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