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# Re: incomplete beta <--> confluent hypergeometric

*From*: Gerard Jungman <jungman at lanl dot gov>
*To*: Peter Schulz-Rittich <rittich at iss dot rwth-aachen dot de>
*Cc*: gsl-discuss at sources dot redhat dot com
*Date*: 19 Sep 2002 13:43:46 -0600
*Subject*: Re: incomplete beta <--> confluent hypergeometric
*References*: <000001c255be$c2bf0140$9101e289@findus>

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