This function relates sequences with common nonzero roots in the denominators of their generating functions. If the generating functions do not have common roots, FAIL is returned.

If the optional parameter y is used, it is the generating function variable for the fifth parameter. The generating function Gy must not involve the variable z in this case.

The nth term of the sequence encoded by Gz (Gy) is expressed as a function of Fn.

The value FAIL is returned if Fz is a trivial rational generating function.

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

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

$\mathrm{Fz}≔\frac{1}{\left(1-az\right)\left(1-bz\right)}\:$

$\mathrm{rgf\_relate}\left(\mathrm{Fz}\,z\,f\left(n\right)\,n\,\frac{1}{1-az}\right)$

$-bf\left(n-1\right)+f\left(n\right)$

$\mathrm{rgf\_relate}\left(\frac{1}{1-bz}\,z\,g\left(n\right)\,n\,\mathrm{Fz}\right)$

$-\frac{bg\left(n\right)}{a-b}+\frac{a{a}^n}{a-b}$

$\mathrm{rgf\_relate}\left(\mathrm{Fz}\,z\,f\left(n\right)\,n\,\frac{1}{\left(1-bw\right)\left(1-cw\right)}\,w\right)$

$-\frac{baf\left(n-1\right)}{b-c}+\frac{bf\left(n\right)}{b-c}-\frac{c{c}^n}{b-c}$

$\mathrm{rgf\_relate}\left(\mathrm{Fz}\,z\,f\left(n\right)\,n\,\frac{1}{{\left(1-az\right)}^2}\right)$

$-b\left(n+1\right)f\left(n-1\right)+\left(n+1\right)f\left(n\right)$

$\mathrm{normal}\left(\right)$

$-nbf\left(n-1\right)-bf\left(n-1\right)+nf\left(n\right)+f\left(n\right)$

$\mathrm{Fz}≔\frac{z}{1-z-z^2}\:$

$\mathrm{rgf\_relate}\left(\mathrm{Fz}\,z\,F\left(n\right)\,n\,{\mathrm{Fz}}^2\right)$

$\left(\frac{n}{5}-\frac{1}{5}\right)F\left(n\right)+\frac{2nF\left(n-1\right)}{5}$

$\mathrm{normal}\left(\right)$

$\frac{F\left(n\right)n}{5}-\frac{F\left(n\right)}{5}+\frac{2nF\left(n-1\right)}{5}$