Re: simplifying a summation / integral
- To: mathgroup at smc.vnet.net
- Subject: [mg64788] Re: simplifying a summation / integral
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Thu, 2 Mar 2006 19:28:14 -0500 (EST)
- Organization: The University of Western Australia
- References: <du6nqk$5ob$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
In article <du6nqk$5ob$1 at smc.vnet.net>, Patrik <hosanagar at gmail.com>
wrote:
> I need a simple closed form expression for:
>
> Sum [(qCr)*(x^r)/r]
> where qCr is q choose r, i.e., (q!/(r!(q-r)!)
> and r is summed from 1 to q
>
> Any thoughts?
Well, try it! Entering
Sum[ Binomial[q, r] x^r / r, {r, 1, q}]
yields
q x HypergeometricPFQ[{1, 1, 1 - q}, {2, 2}, -x]
> I know that the above expression is the same as
> Integral[((1+x)^q - 1)/x]
> So, it'd help if someone can help computing the integral instead.
If you compare
Table[{q, Factor[q x HypergeometricPFQ[{1,1,1-q},{2,2},-x]]}, {q, 5}]
with
Table[{q, Factor[Integrate[((x + 1)^q - 1)/x, x]]}, {q, 5}]
you will see that they are indeed identical.
Essentially your question is whether there is any _simpler_ form. To
answer this question, the excellent book "generatingfunctionology" by
Herbert Wilf is available for free download at
http://www.math.upenn.edu/~wilf/DownldGF.html
Also, the author of the original Mathematica RSolve package was Marko
Petkovsek and, if you are interested, another excellent book, "A=B", by
Marko Petkovsek, Herbert Wilf, and Doron Zeilberger, is also available
for free download at
http://www.cis.upenn.edu/~wilf/AeqB.html
Cheers,
Paul
_______________________________________________________________________
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