New(?) identity for unsigned Stirling Numbers of the first kind
Carlo Wood
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?
-----------------------------
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
Mitch
On Feb 14, 10:15 pm, Carlo Wood <c...@alinoe.com> 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 is not 'new'. See Graham, Knuth, Patashnik, Concrete
Mathematics (2nd ed). Table 265, eq 6.16.
I wouldn't be surprised if it is in Benjamin and Quinn, Proofs that
Really Count, along with a combinatorial proof.
As to the history of this particular identity, I don't know how old it
is.
As to using Mathematica or Maple, wonderful as those aids may be,
just because they don't simplify something doesn't mean it's not easy
(I think both implement the WZ method, but that won't work with
Stirling numbers).
Mitch
acrb...@googlemail.com
> 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.
Carlo Wood
> 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.
Thanks. What about my idea to extend the Stirling numbers
to a function of two real variables? Is this something that
is already been done?
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