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From: Gerry on 27 May 2010 23:25
On May 26, 12:37 pm, Gerry <gerry...(a)gmail.com> wrote:
> On May 25, 10:59 am, Gerry <gerry...(a)gmail.com> wrote:
>
>
>
>
>
> > On May 25, 5:49 am, johnson542 <johnson...(a)verizon.net> wrote:
>
> > > Let k, n be positive integers with k<n.
>
> > > Does any one know the name of the following function:
>
> > > (1/k!) ( [n-k]x^k - C(k,1)[n-k+1]x^{k-1} + ... + (-1)^k [n] )
>
> > > where [n-i] means n(n-1)...(n-k) with the factor n-i deleted.
>
> > > For example, if k=2, n=4, the function is
>
> > > (1/2)*( 12*x^2 -16*x + 6 )
>
> > > I am thinking that this might be some type of hypergeometric function..
>
> > try
>
> >  (-1)^k*n*x^n/(1+k)! *(1-n)_k *B(1/x,n-k,1+k)
>
> > with
>
> >  (n)_k the pochhammer and
> >   B    the incomplete Beta function
>
> > Regards
>
> > Gerry
>
> I have to correct my answer, it should be :
>
>  (-1)^k*n*x^n/k! *(1-n)_k *B(1/x,n-k,1+k)
>
> So that we get for k=5 n=10 that the
>
> quintic
>
>  252x^5-1050x^4+1800x^3-1575x^2+700x-126
>
> is equal to :
>
>  1260x^10 B(1/x,5,6)- Hide quoted text -
>
> - Show quoted text -

So just out of interest we get for the incomplete beta function :

B(1/x;n-k,1+k) = x^n*Sum_{m=0}^k((-1)^m*C(k,m)*x^(k-m)/(n-k+m))

Does anyone else have a reference for this?

I tried to find a reference for this identity but could only find a
relationship with the regularized incomplete beta function I_x(a,b) at
http://en.wikipedia.org/wiki/Beta_function

Maybe i dind't look long enough
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