Math Forum / Mathematics / Research / February 2007

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?

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On a second note, the above allows one to define an extension
w(n,k) over the Reals of |s(n,k)| !

Replace the sum over i with an integral, and the binomial
with Gamma, than it is a convolution product that expresses
w(n+1,k+1) in w(n,k).

w(n+1,k+1) = \integral_{x=0}^{n}
{ w(n,x) \Gamma(x+1)/(\Gamma(k+1)\Gamma(x-k+1)) dx }

Since w(n,x) is expected to be 0 for negative x (or n)
and also when x > n, we might as well integrate from
-\inf till \inf.

w(n+1,k+1) = \integral_{x=-\inf}^{\inf}
{ w(n,x) \Gamma(x+1)/(\Gamma(k+1)\Gamma(x-k+1)) dx }

Then defining

w(n,0) = \delta{n}
w(0,k) = \delta{k}

for real n and k should somehow define w().

For a start, if n=0, we have:

w(1,k+1) = \integral_{x=-\inf}^{\inf}
{ w(0,x) \Gamma(x+1)/(\Gamma(k+1)\Gamma(x-k+1)) dx } =
\Gamma(1)/(\Gamma(k+1)\Gamma(-k+1)) =
1/(\Gamma(1+k)\Gamma(1-k))

Thus w(1,1) = 1, and w(1,k) = 0 for integer values of k \ne 0,
just like |s(1,k+1)|. For non integer values of k we get
both positive and negative values (if you have Mathematica:
Plot[1/(Gamma[1 + k]Gamma[1 - k]), {k, -10, 10}])

In turn, this allows us to find w(2,k) and so on.

I'm not sure how to find non integer values of n yet.
It seems those are all 0, because for 0 < n < 1, we
find:

w(n,k) = \integral_{x=-\inf}^{\inf}
{ w(n-1,x) \Gamma(x+1)/(\Gamma(k)\Gamma(x-k)) dx }

where where w(n-1,x) is expected to be 0. However,
it might very well be that this is only the case for
negative integers, and that w() is non-zero for
fractional values in the area where s() is zero...

Carlo Wood