Re: New(?) identity for unsigned Stirling Numbers of the first kind

On Feb 15, 3:15 am, Carlo Wood <c...@xxxxxxxxxx> wrote:
I discovered the following identity:

Let s(n,k) be the Stirling Numbers of the first kind,
and |s(n,k)| the unsigned Stirling Numbers of the first kind.

Then

\sum_{i=0}^{n} { |s(n,i)| \binom{i}{k} } = |s(n+1,k+1)|

Since neither Mathematica nor Maple were able
to do this simplificiation, I wonder how well known it is?

Is it a "new" identity?

No. It goes back at least as far as (1.4) in

D.S. Mitrinovi\'c, Sur une classe de nombre reli\'es aux
nombres de Stirling, C. R. Acad. Sci. Paris 252 (1961),
2354--2356.

Variations on this theme can be found in Section 2 of

A.C.R. Belton, The monotone Poisson process, in: Quantum
Probability (M. Bo\.zejko, W. M{\l}otkowski and
J. Wysocza\'nski, eds.), Banach Center Publications 73,
Polish Academy of Sciences, Warsaw, 2006

and probably in many other places.

.

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