On 12/31/06, Michel Van den Bergh <michel.vandenbergh at uhasselt.be> wrote:
> >
> > >/
> > />/ About computing the sum of rational functions by polygamma functions:
> > />/ first this needs to compute the roots of the denominator --> a numerical method,
> > />/ so the obtained precision is not that os polygamma but that of the numerical
> > />/ method for computing the roots of the denominator.
> > />/
> > />/
> > />/
> > /Maxima can work with algebraic integers! tellrat is the command.
> >
> >
>
> And a common subcase would be where the roots are half integers. In that
> case Maxima can use
> simplication to obtain the symbolic result in terms of the zeta function.
I will soon commit a new version of closed_form (along with some
bugfixes for polygamma in maxima) to cvs. It will convert sums of
rational functions to psi functions. Maxima knows enough about psi to
simplify to zeta and other functions when it is possible. Here's what
I have currently:
(%i2) sum(1/n/(2*n+1), n, 1, inf)$
(%i3) closed_form(%);
(%o3) 2-2*log(2)
(%i4) sum(n/(3*n+1)/(2*n+1)^2, n, 1, inf)$
(%i5) closed_form(%), ratsimp;
(%o5) -(12*sqrt(3)*log(3)-16*sqrt(3)*log(2)-sqrt(3)*%pi^2+4*%pi)/(8*sqrt(3))
Andrej