[seqfan] A formula involving Stirling-numbers 1 kind, which seems to be zero for all parameters

Gottfried Helms helms at uni-kassel.de
Sat Dec 31 03:19:33 CET 2011

Dear seqfans -

 I've discussed this problem already in another forum but could not
 get a solution for the general case.

 Consider the following formula for parameters r and c , (r,c >=0 )
 which involves the Stirlingnumbers first kind and binomials:


           inf          s1(k+1, k-r+1)
su(r,c) = sum  (-1)^k ---------------- * (-1)^c * (1+c)^k * binom(k+1,c+1)
           k=a                k!

           where   a = max(r,c)

Hypothesis:    su(r,c) = 0  for all  r,c>=0


Empirically these sums are all tending to zero as the index k increases
unboundedly, but I have no idea, how I could prove this in that
generality for all indexes r and c.
I used Mathematica (in its limited edition online at wolframalpha) for
a couple of instances of r and c; and for r=0 and r=1 mathematica could
even show, that the result is zero independently of the other parameter c.
However, for bigger r the description of the stirling-numbers is too
complicated - but note, that the matrix of Stirlingnumbers is evaluated
along its subdiagonals!

Here is a short procedure in Pari/GP; I have programmed the computation
of the S1-matrix, but if I've got this correctly the new versions of Pari/GP
have them readily implemented.

n=512;S1 = makemat_S1(n);   \\ prepare array of 512 x 512

stir1(r,c)=S1[1+r,1+c]   \\ formal procedure to provide Stirling-numbers with zero-based index
                         \\ may be unneccessary in new versions of Pari/GP

{ su(r,c)=local(a=max(r,c),su);
  su=sum(k=a, n-2,
             (-1)^k * stir1(k+1,k-r+1) / k!
           * (-1)^c * (1+c)^k * binomial(k+1,c+1)
 return(su); }

su(33,33)*1.0   \\ some example

Unfortunately the form of the series seem to be similar to that of the
exponential-function, where the partial sums may grow to huge values
before they decrease (and converge) to their final value, so with
dimension n=512 we may ask for su(r,c) in the range of r,c <30 or <40
for suggestive results.

Does someone around here have an idea how to prove (or disprove, it
my hypothesis happens to be wrong...) that heuristic?

Thanks in advance -
and a happy new year to all seq-fan-tastics -

Gottfried Helms

Further references :

A link to an article, where I describe where the problem comes from
and what I have so far:
[Helms2011] http://go.helms-net.de/math/divers/inverseNullmatrix.pdf

The following is a related link into the OEIS:
[OEIS]      Triangle from inverse scaled Pochhammer symbols.
            Wolfdieter Lang, Sep 12, 2005
            in "Online encyclopedia of Integer sequences"(OEIS) maintained by N.J.A.Sloane

Here is a discussion in MSE
[MSE1112]   Does the inverse of this matrix of size n times n go to the zero matrix?
            Gottfried Helms, 12'2011
            question in math.stackexchange.com, as a spinoff of [MSE1101]

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