 rgf_relate - Maple Help

genfunc

 rgf_relate
 relate sequences with common factors in their generating functions Calling Sequence rgf_relate(Fz, z, Fn, n, Gz) rgf_relate(Fz, z, Fn, n, Gy, y) Parameters

 Fz - rational generating function z - name, generating function variable Fn - expression for nth term of the sequence encoded by Fz n - name, index variable for Fn Gz, Gy - rational generating function y - (optional) name, generating function variable for Gy Description

 • 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. Examples

 > $\mathrm{with}\left(\mathrm{genfunc}\right):$
 > $\mathrm{Fz}≔\frac{1}{\left(1-az\right)\left(1-bz\right)}:$
 > $\mathrm{rgf_relate}\left(\mathrm{Fz},z,f\left(n\right),n,\frac{1}{1-az}\right)$
 ${-}{b}{}{f}{}\left({n}{-}{1}\right){+}{f}{}\left({n}\right)$ (1)
 > $\mathrm{rgf_relate}\left(\frac{1}{1-bz},z,g\left(n\right),n,\mathrm{Fz}\right)$
 ${-}\frac{{b}{}{g}{}\left({n}\right)}{{a}{-}{b}}{+}\frac{{a}{}{{a}}^{{n}}}{{a}{-}{b}}$ (2)
 > $\mathrm{rgf_relate}\left(\mathrm{Fz},z,f\left(n\right),n,\frac{1}{\left(1-bw\right)\left(1-cw\right)},w\right)$
 ${-}\frac{{b}{}{a}{}{f}{}\left({n}{-}{1}\right)}{{b}{-}{c}}{+}\frac{{b}{}{f}{}\left({n}\right)}{{b}{-}{c}}{-}\frac{{c}{}{{c}}^{{n}}}{{b}{-}{c}}$ (3)
 > $\mathrm{rgf_relate}\left(\mathrm{Fz},z,f\left(n\right),n,\frac{1}{{\left(1-az\right)}^{2}}\right)$
 ${-}{b}{}\left({n}{+}{1}\right){}{f}{}\left({n}{-}{1}\right){+}\left({n}{+}{1}\right){}{f}{}\left({n}\right)$ (4)
 > $\mathrm{normal}\left(\right)$
 ${-}{n}{}{b}{}{f}{}\left({n}{-}{1}\right){-}{b}{}{f}{}\left({n}{-}{1}\right){+}{n}{}{f}{}\left({n}\right){+}{f}{}\left({n}\right)$ (5)
 > $\mathrm{Fz}≔\frac{z}{1-z-{z}^{2}}:$
 > $\mathrm{rgf_relate}\left(\mathrm{Fz},z,F\left(n\right),n,{\mathrm{Fz}}^{2}\right)$
 $\left(\frac{{n}}{{5}}{-}\frac{{1}}{{5}}\right){}{F}{}\left({n}\right){+}\frac{{2}{}{n}{}{F}{}\left({n}{-}{1}\right)}{{5}}$ (6)
 > $\mathrm{normal}\left(\right)$
 $\frac{{F}{}\left({n}\right){}{n}}{{5}}{-}\frac{{F}{}\left({n}\right)}{{5}}{+}\frac{{2}{}{n}{}{F}{}\left({n}{-}{1}\right)}{{5}}$ (7)