On Thu, 21 Aug 2008 01:45:32 -0700
"William Stein" <[EMAIL PROTECTED]> wrote:
>
> On Thu, Aug 21, 2008 at 1:29 AM, Alec Mihailovs <[EMAIL PROTECTED]> wrote:
> >
> > Both Maple and Mathematica give wrong answers to the following sum,
> >
> > Sum[Binomial[n, k]/Binomial[2 n, k]/k! (2 x)^k, {k, 0, n}]
> >
> > I tried to find a way to calculate it in SAGE, but couldn't find symbolic
> > sums in the documentation. Is Maxima supposed to be used directly?
> >
>
> Nobody wrote a wrapper for symbolic sums in Sage so you have to
> use maxima directly for that.
I haven't actually tried it, but I think this example is beyond maxima
as well.
MMA seems to just perform a rewrite rule, and hide the sum inside the
BesselI. The answer given by MMA satisfies the same recurrence
-x^2 SUM[n]-(1+2 n) (3+2 n) SUM[1+n]+(1+2 n) (3+2 n) SUM[2+n] == 0
found by the implementation of Zeilberger's algorithm here:
http://www.risc.uni-linz.ac.at/research/combinat/software/PauleSchorn/index.php
However, the initial values don't match. :)
If you plug in the above recurrence into Petkovsek's Hyper, which is
available here:
http://www.fmf.uni-lj.si/~petkovsek/distrib.m
it doesn't give an answer. This means that your expression doesn't have
a hypergeometric closed form in the sense of A=B, p. 143 [1]:
http://www.cis.upenn.edu/~wilf/AeqB.html
Sage will have symbolic summation capabilities soon, but I doubt if it
will ever be able to simplify this expression.
What are your expectations from a "correct answer"? How did you come
across this example? If you know/conjecture a simpler form for this
expression, you can prove that they are equal by showing that they both
satisfy the above recurrence, and agree on the inital values.
Cheers,
Burcin
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