[Help-gsl] Re: Incomplete Beta function for negative b ?

[Lionel B]

Rodney Sparapani wrote:
> Lionel B wrote:
> > Greetings,
> >
> > I need values of gsl_sf_beta_inc (double a, double b, double x) for 0 
> > < a < 1, -1 < b < 0. The incomplete Beta function is, as far as I 
> > know, well-defined in this case.
> >
> > I'm a bit stumped - haven't managed to find any identities involving 
> > B(z;a,b) that work for me here...
>
> Hi Lionel:
>
> I stared at that for way too long before I tried it with Mathematica.
> It turns out that the result is based on Hypergeometric 2F1 which are
> available in GSL as well.

Hi,

Right, that's the way I'd programmed it originally, as it happens (and 
sure, it works)... I guess I just wanted to use the incomplete Beta as 
it is in some sense a "simpler" function than the hypergeometric - at 
least in the sense that it takes fewer arguments.

As it happens, I think I've cracked it via the following identities: 
firstly:

B(a,b;z) = B(a,b) - B(b,a;1-z)

to swap the negative b into the first argument position (the negative 
argument in B(a,b) is not a problem as it is simple to derive an 
equivalent expression in terms of positive arguments using elementary 
properties of the Gamma function).

Secondly:

a B(a,b;z) = z^a (1-z)^b + (a+b) B(a+1,b;z)

allows to add one to 1 to the first (now negative) argument which in my 
case suffices to yield all positive arguments (sorry if that's a bit 
convoluted).

Lionel