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

 From: Lionel B Subject: [Help-gsl] Re: Incomplete Beta function for negative b ? Date: Tue, 29 Aug 2006 22:44:23 +0100 User-agent: Thunderbird 1.5 (Windows/20051201)

```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

```