The function finds the closed form expansion of the nth term in the sequence encoded by the rational generating function Fz.

If the denominator of Fz has polynomial factors of degree greater than two that cannot be factored over the rationals, an inert Sum expression involving RootOf values is returned.

The global variables _J and _R are used in the RootOf expressions.

The command with(genfunc,rgf_expand) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{genfunc}\right)\:$

$\mathrm{rgf\_expand}\left(\frac{1}{1-5z+6z^2}\,z\,n\right)$

$-22^n+33^n$

$\mathrm{rgf\_expand}\left(\frac{y}{1-y-y^2}\,y\,k\right)$

$-\frac{\left(\sqrt{5}-1\right)\sqrt{5}{\left(-\frac{2}{-\sqrt{5}+1}\right)}^k}{5\left(-\sqrt{5}+1\right)}-\frac{\sqrt{5}{\left(-\frac{2}{\sqrt{5}+1}\right)}^k}{5}$

$\mathrm{rgf\_expand}\left(\frac{1+2z+3z^2}{1-z-z^2-z^3}\,z\,n\right)$

$\sum _{\mathrm{\_R}=\mathrm{RootOf}\left({\mathrm{\_Z}}^3+{\mathrm{\_Z}}^2+\mathrm{\_Z}-1\right)}\left(-\frac{\left(\underset{z\to \mathrm{\_R}}{lim}\left(-\frac{\left(z-\mathrm{\_R}\right)\left(3z^2+2z+1\right)}{z^3+z^2+z-1}\right)\right){\left(\frac{1}{\mathrm{\_R}}\right)}^n}{\mathrm{\_R}}\right)$