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 _______________________________________________________________________ Paul Abbott Phone: 61 8 6488 2734 School of Physics, M013 Fax: +61 8 6488 1014 The University of Western Australia (CRICOS Provider No 00126G) AUSTRALIA http://physics.uwa.edu.au/~paul