incomplete beta <--> confluent hypergeometric

[Gerard Jungman]

On Fri, 2002-09-06 at 10:02, Peter Schulz-Rittich wrote:
> Hi,
>  
> I ran into this posting searching for a relation between the incomplete
> beta function and the confluent hypergeometric function. It seems that
> the latter could be expressed as an infinite sum over the former .? Do
> you have a hint? Or a reference?
>  
> Thanks a lot in advance!

Hi. Took me awhile to get to this message. Sorry. You've probably
figured it out already. There is a direct relation to the usual
2F1 hypergeometric function; this is probably what I had in mind:

with definition:

  B_x(p,q) = \int_0^x  t^{p-1} (1-t)^{q-1} dt

we have:

  B_x(p, q) = x^p p^{-1}  F(p, 1-q; p+1; x)
            = x^p (1-x)^{q-1} p^{-1} F(1, 1-q; p+1; x/(x-1))
            = x^p (1-x)^q p^{-1} F(p+q, 1; p+1; x)

source: N. Temme, Special Functions

The various relations follow from standard hypergeometric
identities, so there are other forms that might be useful
as well.

--
G. Jungman