asking references
Edwin Clark
eclark at math.usf.edu
Wed Mar 26 23:02:34 CET 2003
On Wed, 26 Mar 2003, Emeric Deutsch wrote:
> Seqfans,
> I believe that the easily proved formula
> sum(n!/k!, k=0..n) = [n!e]
> is given in some form in many calculus books. I'd like to ask
> the favor of some references. Maple gives eGAMMA(n+1,1) with
> a boldfaced e, whatever that means.
> Thanks,
> Emeric
> [] is the floor function.
>
>
In Maple's answer the boldfaced e is exp(1), the same
as the e in your formula. The definition of GAMMA(n+1,1)
is given in the help page for GAMMA:
-----------------------------------------------------------------------
The incomplete Gamma function is defined as:
GAMMA(a,z) = GAMMA(a) - z^a/a 1F1(a,1+a,-z)
where 1F1 is the confluent hypergeometric function (in Maple notation,
1F1(a,1+a,-z) = hypergeom([a],[1+a],-z)).
For Re(a)>0, we also have the integral representation
GAMMA(a,z) = int( exp(-t)*t^(a-1), t=z..infinity )
----------------------------------------------------------------------
--Edwin
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