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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