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 $(\exp(x)-1)^c$.

extended matrix of scaled Stirlingnumbers 2nd kind

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).

extended matrix of scaled Stirlingnumbers 2nd kind

Note, this is a specific detail in the same area as in my earlier question here

  • $\begingroup$ Have you seen this? Formula 2.4 there might be of interest to you. $\endgroup$ – J. M. isn't a mathematician Dec 12 '15 at 8:30
  • $\begingroup$ @j.m.: Yes, I know it, thanks! It's long ago that I read it but with D. Knuth's articles it's always worth to read into them again... Yes, the transposed relation is a common effect with many matrices - a simpler example is the Pascalmatrix which can similarly be extended (with its own inverse). In a closely related question in MO I ask for the relation between the "vertical" and "horizontal" generating functions (because I had a couple of interesting heuristiscs) and someone was able to show the systematic relation (I'll add the link when I have it) $\endgroup$ – Gottfried Helms Dec 12 '15 at 9:55
  • $\begingroup$ So, doesn't that reflection relation answer the question in your title? $\endgroup$ – J. M. isn't a mathematician Dec 12 '15 at 10:20
  • $\begingroup$ @j.m.: the generalization is a bit different; in my concept above the Stirling numbers first kind are cofactored with factorials while this is not so with Knuth's arrangement. And the article to which I linked in my own answer there is even another generalization - so I do not yet know the most opportune one. My main interest were in any case that numbers of the bottom-left quadrant (or say better : "formal symbols" because they only "enter reality" when the gamma(0) term is removed - which is somehow fiddling and needs much more consideration of why and when this is meaningful... $\endgroup$ – Gottfried Helms Dec 12 '15 at 11:08

There is a fairly involved discussion in

On Stirling numbers for complex arguments and Hankel contours
Philippe Flajolet, Helmut Prodinger Rapport de recherche, No 3373, Mar 1998
ISSN 0249-6399

They start with the same logic via the generating functions $(\exp(x)-1)^c$, they even generalize to complex indices in row and column. Unfortunately I can't follow that specific explanations by "Hankel-contour-integrals" yet.


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