[Maxima] simplification of products of gamma functions & bfloat bug

[Dieter Kaiser]

Am Sonntag, den 22.02.2009, 18:50 -0600 schrieb Barton Willis: 
> Is there a Maxima function that simplifies (%o47) to 1? The composition
> minfactorial(makefact(...)) doesn't simplify (%o47) to 1. Also, maybe it
> has already been fixed, but (%o48) shows a bug:
> 
>  (%o47) (gamma(1/7)*gamma(4/21)*gamma(17/21)*gamma(6/7)-gamma(4/21)*gamma
>  (10/21)*gamma(11/21)*gamma(17/21))/(gamma(1/7)*gamma(10/21)*gamma
>  (11/21)*gamma(6/7))

The Maxima functions I know can not simplify products of gamma
functions. Perhaps we can implement some rules for the product of gamma
functions. These are some examples

(1) gamma(z)*gamma(w) = factorial(z+w-2)/binomial(w+z-2,z-1)
(2) gamma(z)*gamma(w) = gamma(z+w)*beta(z,w)

(3) gamma(z)/gamma(w) = factorial(z-w)/binomial(z-1,z-w)
(4) gamma(z)/gamma(w) = pochhammer(w,z-w)

But the first rule will not work in the example above, because we get an
undefined factorial(-1). The second rule will simplify to an expression
with the sin function.

Dieter Kaiser