I have the following set of series involving the Stirling numbers 1'st kind and binomials, which can be understood as a set of dot-products of row- and column-vectors of two infinite matrices (where *R* and *C* indicate rows and columns, beginning at zero):

$$ w_{R,C} =\sum_{k=\max(R,C)}^\infty (-1)^k {s_1(1+k,1+k-R)\over k!} \cdot (-1)^C (1+C)^k \cdot \binom {1+k}{1+C} $$

I've tested this heuristically for several *R* and *C* and always approximated zero; also wolfram-alpha can evaluate this explicitely to zero if feeded with

sum (-1)^k * StirlingS1(k+1,1+k-R)/k! * (1+C)^k * binomial(1+k,1+C), for k=max(C,R) to infty

where we replace $C$, $R$ and $\max(C,R)$ with actual values.

However, I've no option to let wolfram-alpha answer this in general.

I've proved this for $C=0,1,2$ and the first few $R$ using exponential generating functions, but again, a general proof is out of reach for me (possibly I'm overlooking something trivial like telescoping...), so I ask for help here.

The convention for Stirling numbers first kind as in Math'ica, indexes beginning at zero:

$ \small \qquad \qquad \begin{array} {rrrrr} 1 & . & . & . & . & . \\\ 0 & 1 & . & . & . & . \\\ 0 & -1 & 1 & . & . & . \\\ 0 & 2 & -3 & 1 & . & . \\\ 0 & -6 & 11 & -6 & 1 & . \\\ 0 & 24 & -50 & 35 & -10 & 1 \end{array} $

If some background is of interest: here are the questions on MSE

https://math.stackexchange.com/questions/16228 // question of some user which motivated me to look at an example

https://math.stackexchange.com/questions/89853 // my follow-up question dealing with the current problem

and a more worked out treatize on this in a pdf-file http://go.helms-net.de/math/divers/InverseNullmatrix.pdf

** [update]** Hmm, after 1 1/2 years I've looked at the question again and still do not have an idea how to construct a proof for the whole set of identities. To possibly stimulate helpful answers here I'll insert pictures of the matrices - perhaps it helps to get an immediate idea when the patterns are more visible/obvious than in the bare formula above.

This is (the top-left-segment of) the matrix $M$ in question.

This are the **L** and **D** factors of the L D U-decomposition. Because it seems convenient to recognize familiar numbers I've documented the product **LD** = **L** **D**

This is the **U** factor:

This is the reciprocal of **U** (call it **UI**):

This is the reciprocal of **LD** (call it **LDI**):

and in the limit for infinite size of the **UI** and **LDI**, the product **UI** * **LDI** = **MI** = **0** by hypothese.

Here are the matrices UI and LDI in a near-symbolic display, the coefficients $s1[r,c]$ are the Stirling numbers first kind.

Reformulating the dotproducts using their exponential generating functions it is not difficult to prove the identities for a couple of examples.

But what is missing is the proof for the

**.**

*full set of dotproducts*

*[/update]*