A binomial sum

[Edwin Clark]

Allowing Maple to think for me, generally

product(a*j+b,j=1..n) = 

          a^n*GAMMA((a*n+a+b)/a)/GAMMA((a+b)/a)

And also Maple gives:

product(b*j^2+c*j+d,j=1..n);

   (n + 1)                   -c + %1                    -c - %1    /
  b        GAMMA(n + 1 - 1/2 -------) GAMMA(n + 1 - 1/2 -------)  /
                                b                          b     /

        /                -c + %1                -c - %1 \
        |b GAMMA(1 - 1/2 -------) GAMMA(1 - 1/2 -------)|
        \                   b                      b    /

              2
  %1 := sqrt(c  - 4 b d)


But it has no answer for  product(a*j^3+b*j^2+c*j+d,j=1..n).

--Edwin


On Tue, 20 Apr 2004, Karol PENSON wrote:

> 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
> > 
> 
> 

-- 
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    W. Edwin Clark, Math Dept, University of South Florida,
           http://www.math.usf.edu/~eclark/
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