Re: New(?) identity for unsigned Stirling Numbers of the first kind
- From: "Mitch" <maharri@xxxxxxxxx>
- Date: 15 Feb 2007 11:04:22 -0500
On Feb 14, 10:15 pm, 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 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
.
- References:
- New(?) identity for unsigned Stirling Numbers of the first kind
- From: Carlo Wood
- New(?) identity for unsigned Stirling Numbers of the first kind
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