The RationalCanonicalForm[1](F,x) and RationalCanonicalForm[2](F,x) functions construct the first and second rational canonical forms for F, where F is a rational function in x over a field of characteristic $0$, respectively.

If RationalCanonicalForm is called without any indexing, the first rational canonical form is used.

A sequence of five elements $z,r,s,u,v$, where z is an element in K and $r,s,u,v$ are monic polynomials over K such that the following three conditions are satisfied, is returned:

$F=\frac{zrE\left(\frac{u}{v}\right)v}{su}$.

$\gcd \left(r\,E^{k\left(s\right)}\right)=1$ for all integers k.

$\gcd \left(r\,u·E\left(v\right)\right)=1$, $\gcd \left(s\,E\left(u\right)·v\right)=1$.

Note: E is the automorphism of K(x) defined by $E\left(f\left(x\right)\right)=f\left(x+1\right)$.

The five-tuple $z,r,s,u,v$ that satisfies the three conditions is a strict rational normal form for F. It is a normal form, not a canonical form. See the References section for information about definitions and constructions of the first and second rational canonical forms.

This function is part of the RationalNormalForms package, and so it can be used in the form RationalCanonicalForm(..) only after executing the command with(RationalNormalForms). However, it can always be accessed through the long form of the command by using RationalNormalForms[RationalCanonicalForm](..).

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

$F≔\frac{\frac{3}{2}n\left(n+2\right)\left(3n+2\right)\left(3n+4\right)}{\left(n-1\right)\left(2n+9\right){\left(n+4\right)}^2}$

$F≔\frac{3n\left(n+2\right)\left(3n+2\right)\left(3n+4\right)}{2\left(n-1\right)\left(2n+9\right){\left(n+4\right)}^2}$

$\mathrm{z1},\mathrm{r1},\mathrm{s1},\mathrm{u1},\mathrm{v1}≔\mathrm{RationalCanonicalForm}\left[1\right]\left(F\,n\right)$

$\mathrm{z1},\mathrm{r1},\mathrm{s1},\mathrm{u1},\mathrm{v1}≔\frac{27}{4},\left(n+\frac{2}{3}\right)\left(n+\frac{4}{3}\right),\left(n+\frac{9}{2}\right)\left(n+4\right),n-1,\left(n+3\right)\left(n+2\right)$

$\mathrm{z2},\mathrm{r2},\mathrm{s2},\mathrm{u2},\mathrm{v2}≔\mathrm{RationalCanonicalForm}\left[2\right]\left(F\,n\right)$

$\mathrm{z2},\mathrm{r2},\mathrm{s2},\mathrm{u2},\mathrm{v2}≔\frac{27}{4},\left(n+\frac{2}{3}\right)\left(n+\frac{4}{3}\right),\left(n-1\right)\left(n+\frac{9}{2}\right),1,{\left(n+3\right)}^2{\left(n+2\right)}^2\left(n+1\right)n$

$\mathrm{evalb}\left(F=\mathrm{normal}\left(\frac{\mathrm{z1}\left(\frac{\mathrm{r1}}{\mathrm{s1}}\right)\mathrm{subs}\left(n=n+1\,\frac{\mathrm{u1}}{\mathrm{v1}}\right)}{\frac{\mathrm{u1}}{\mathrm{v1}}}\right)\right)$

$\mathrm{true}$

$\mathrm{LREtools}\left[\mathrm{dispersion}\right]\left(\mathrm{r1}\,\mathrm{s1}\,n\right),\mathrm{LREtools}\left[\mathrm{dispersion}\right]\left(\mathrm{s1}\,\mathrm{r1}\,n\right)$

$\mathrm{FAIL},\mathrm{FAIL}$

$\gcd \left(\mathrm{r1}\,u\mathrm{subs}\left(n=n+1\,\mathrm{v1}\right)\right),\gcd \left(s\,\mathrm{subs}\left(n=n+1\,\mathrm{u1}\right)v\right)$

$1,1$

Abramov, S., and Petkovsek, M. "Canonical Representations of Hypergeometric Terms." FPSAC'01. 2000.