A binomial sum

[Karol PENSON]

The third product in Henry's mail can be expressed with the gamma 
function. Since the gamma function is as common tool as factorial
I still believe that the second line below is a closed form of the first. 


simplify(product(3*j-2,j=1..n));
                    (n + 1/2)
                   3          GAMMA(n + 1/3) GAMMA(2/3)
               1/2 ------------------------------------
                                    Pi

Cheers,
 Karol A. Penson

On Mon, 19 Apr 2004, Henry Gould wrote:

> Date: Mon, 19 Apr 2004 17:16:38 -0400
> From: Henry Gould <gould at math.wvu.edu>
> To: Sequence Fans Mailing List <seqfan at ext.jussieu.fr>
> Subject: Re: A binomial sum 
> 
> Gentlemen -
> If we tabulate ANY function, as e.g. a Table of Prime numbers, or admit a
> Hypergeometric function as a tabulated function, then of course we have a
> "closed form".
> We can express the product of the even nunbers 2,4,6,...2n by the "closed
> form" (2^n)n!.
> We can express the product of the odd numbers 1,3,5,...2n-1 by the "closed
> form" (2n)!/(2^n)n!,
> but we cannot do the same with the product of 1,4,7,10,13,16,...,3n-2.  Of
> course if we admit another this as a tabulated function and assign it a
> name, e.g. P(n,3), then we ipso facto have a "formula."
> It is all relative to what we accept as an "elementary" function. My point
> is that the sum in the query cannot be done with simple factorial products
> and ratios.
> 
> Cheers and Salud!
> 
> Henry
> 

-- 
_________________________________________________________________________
Karol A. PENSON
Universite Paris 6              |  Internet : penson at lptl.jussieu.fr.
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