Re: A closed form challenge!

TP2006 wrote:


Is there a closed form for the coefficient of x^k in the product

(x+n)(2x+n-1)(3x+n-2)...((n-1)x+2),

where n is a fixed positive integer?

The way to attack this problem seems to me the following:
(x+n)*(2*x+n-1)*(3*x+n-2)*...*((n-1)*x+2) =
(x+n)*(y+1)*(y+2)*...*(y+n-2)*(x-1)^(n-2)
with y = (x+n)/(x-1)
As you can see now we are in the domain of the expansion of the gamma
function which gives, depending on your background, harmonic numbers or
harmonic sums. On the whole neither mathematica nor maple are very good
with these sums. They are used nowadays a bit more commonly in
Quantum Field Theory, especially since the invention of QCD. They have
undergone quite some transformations since then and a good paper to start
on them is to go to this link:
http://www.slac.stanford.edu/spires/find/hep/www?key=3765792
Then you can have a look at papers that refer to it to get to the more
recent literature.
http://www.slac.stanford.edu/spires/find/hep?c=IMPAE,A14,2037
There exist programs in FORM (have a look at http://www.nikhef.nl/~form
and in the packages you can find the summer package) for whole categories
of sums over these objects, including the sums that you will need to get
terms in your expansion (when you expand the y into x, you will get a sum
over the harmonic sums. Another sum comes from the (x-1)^(n-2). Both
should be in the range of what summer can do).
Recently the two Smirnovs (V.A. and A.V.) have reprogrammed the summer
package
(and probably added to it) into a mathematica package (the son converted it
for the father so that he can use it for his physics work).

I hope this helps you a bit on your way.

Jos Vermaseren


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