Let F be a rational function of n over a field K of characteristic 0. The RationalCanonicalForm[i](F,n) calling sequence constructs the ith rational canonical forms for F, $i=\left\{1\,2\,3\,4\right\}$.

If the RationalCanonicalForm command is called without an index, the first rational canonical form is constructed.

The output is a sequence of 5 elements $z,r,s,u,v$, called $\mathrm{RNF}\left(F\right)$, where z is an element of K, and $r,s,u,v$ are monic polynomials over K such that:

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

$\gcd \left(r\,E^{k\left(s\right)}\right)=1$ 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(n) defined by $E\left(F\left(n\right)\right)=F\left(n+1\right)$.

The five-tuple $z,r,s,u,v$ that satisfies the three conditions is a strict rational normal form for F. The rational functions $\frac{zr}{s}$ and $\frac{u}{v}$ are called the kernel and the shell of an $\mathrm{RNF}\left(F\right)$, respectively.

Let $\mathrm{\phi }=\left(z\,r\,s\,u\,v\right)$ be any RNF of a rational function F. Then the degrees of the polynomials r and s are unique, and have minimal possible values in the sense that if $F\left(n\right)=\frac{p\left(n\right)E\left(G\left(n\right)\right)}{q\left(n\right)G\left(n\right)}$ where p, q are polynomials in n, and G is a rational function of n, then $\mathrm{degree}\left(r\right)\le \mathrm{degree}\left(p\right)$ and $\mathrm{degree}\left(s\right)\le \mathrm{degree}\left(q\right)$.

If $i=1$ then $\mathrm{degree}\left(v\right)$ is minimal.

If $i=2$ then $\mathrm{degree}\left(u\right)$ is minimal.

If $i=3$ then $\mathrm{degree}\left(u\right)+\mathrm{degree}\left(v\right)$ is minimal, and under this condition, $\mathrm{degree}\left(v\right)$ is minimal.

If $i=4$ then $\mathrm{degree}\left(u\right)+\mathrm{degree}\left(v\right)$ is minimal, and under this condition, $\mathrm{degree}\left(u\right)$ is minimal.

$\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{Hypergeometric}\right]\right)\:$

$\mathrm{\nu }≔n\left(n+2\right)\left(n-4+\mathrm{sqrt}\left(2\right)\right)\left(n-3+\mathrm{sqrt}\left(2\right)\right)\left(n+2+\mathrm{sqrt}\left(2\right)\right)\left(n+11+\mathrm{sqrt}\left(2\right)\right)$

$\mathrm{\nu }≔n\left(n+2\right)\left(n-4+\sqrt{2}\right)\left(n-3+\sqrt{2}\right)\left(n+2+\sqrt{2}\right)\left(n+11+\sqrt{2}\right)$

$\mathrm{de}≔\left(n-3\right){\left(n-2\right)}^2\left(n+6\right)\left(n+12\right)\left(n-1+\mathrm{sqrt}\left(2\right)\right)\left(n+1+\mathrm{sqrt}\left(2\right)\right)$

$\mathrm{de}≔\left(n-3\right){\left(n-2\right)}^2\left(n+6\right)\left(n+12\right)\left(n-1+\sqrt{2}\right)\left(n+1+\sqrt{2}\right)$

$F≔\frac{\mathrm{\nu }}{\mathrm{de}}$

$F≔\frac{n\left(n+2\right)\left(n-4+\sqrt{2}\right)\left(n-3+\sqrt{2}\right)\left(n+2+\sqrt{2}\right)\left(n+11+\sqrt{2}\right)}{\left(n-3\right){\left(n-2\right)}^2\left(n+6\right)\left(n+12\right)\left(n-1+\sqrt{2}\right)\left(n+1+\sqrt{2}\right)}$

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

$\mathrm{z1},\mathrm{r1},\mathrm{s1},\mathrm{u1},\mathrm{v1}≔1,\left(n-4+\sqrt{2}\right)\left(n-3+\sqrt{2}\right),\left(n-3\right)\left(n+6\right)\left(n+12\right),{\left(n+1+\sqrt{2}\right)}^2{\left(n-1\right)}^2{\left(n-2\right)}^2\left(n+1\right)n\left(n+10+\sqrt{2}\right)\left(n+9+\sqrt{2}\right)\left(n+8+\sqrt{2}\right)\left(n+7+\sqrt{2}\right)\left(n+6+\sqrt{2}\right)\left(n+5+\sqrt{2}\right)\left(n+4+\sqrt{2}\right)\left(n+3+\sqrt{2}\right)\left(n+2+\sqrt{2}\right)\left(n+\sqrt{2}\right)\left(n-1+\sqrt{2}\right),1$

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

$\mathrm{z2},\mathrm{r2},\mathrm{s2},\mathrm{u2},\mathrm{v2}≔1,\left(n+2+\sqrt{2}\right)\left(n+11+\sqrt{2}\right),\left(n-3\right){\left(n-2\right)}^2,1,{\left(n-2+\sqrt{2}\right)}^2{\left(n-3+\sqrt{2}\right)}^2{\left(n+5\right)}^2{\left(n+4\right)}^2{\left(n+3\right)}^2{\left(n+2\right)}^2\left(n+\sqrt{2}\right)\left(n-1+\sqrt{2}\right)\left(n-4+\sqrt{2}\right)\left(n+11\right)\left(n+10\right)\left(n+9\right)\left(n+8\right)\left(n+7\right)\left(n+6\right)\left(n+1\right)n$

$\mathrm{z3},\mathrm{r3},\mathrm{s3},\mathrm{u3},\mathrm{v3}≔\mathrm{RationalCanonicalForm}\left[3\right]\left(F\,n\right)$

$\mathrm{z3},\mathrm{r3},\mathrm{s3},\mathrm{u3},\mathrm{v3}≔1,\left(n-4+\sqrt{2}\right)\left(n+11+\sqrt{2}\right),\left(n-3\right)\left(n+6\right)\left(n+12\right),\left(n+1+\sqrt{2}\right){\left(n-1\right)}^2{\left(n-2\right)}^2\left(n+1\right)n,\left(n-2+\sqrt{2}\right)\left(n-3+\sqrt{2}\right)$

$\mathrm{z4},\mathrm{r4},\mathrm{s4},\mathrm{u4},\mathrm{v4}≔\mathrm{RationalCanonicalForm}\left[4\right]\left(F\,n\right)$

$\mathrm{z4},\mathrm{r4},\mathrm{s4},\mathrm{u4},\mathrm{v4}≔1,\left(n-4+\sqrt{2}\right)\left(n+11+\sqrt{2}\right),\left(n-3\right)\left(n-2\right)\left(n+12\right),\left(n-1\right)\left(n-2\right)\left(n+1+\sqrt{2}\right),\left(n+5\right)\left(n+4\right)\left(n+3\right)\left(n+2\right)\left(n-2+\sqrt{2}\right)\left(n-3+\sqrt{2}\right)$

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

$\mathrm{true}$

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

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

$\gcd \left(\mathrm{r1}\,\mathrm{u1}\mathrm{subs}\left(n=n+1\,\mathrm{v1}\right)\right),\gcd \left(\mathrm{s1}\,\mathrm{subs}\left(n=n+1\,\mathrm{u1}\right)\mathrm{v1}\right)$

$1,1$

$\mathrm{degree}\left(\mathrm{r1}\,n\right),\mathrm{degree}\left(\mathrm{r2}\,n\right),\mathrm{degree}\left(\mathrm{r3}\,n\right),\mathrm{degree}\left(\mathrm{r4}\,n\right)$

$2,2,2,2$

$\mathrm{degree}\left(\mathrm{s1}\,n\right),\mathrm{degree}\left(\mathrm{s2}\,n\right),\mathrm{degree}\left(\mathrm{s3}\,n\right),\mathrm{degree}\left(\mathrm{s4}\,n\right)$

$3,3,3,3$

$\mathrm{degree}\left(\mathrm{v1}\,n\right),\mathrm{degree}\left(\mathrm{v2}\,n\right),\mathrm{degree}\left(\mathrm{v3}\,n\right),\mathrm{degree}\left(\mathrm{v4}\,n\right)$

$0,23,2,6$

$\mathrm{degree}\left(\mathrm{u1}\,n\right),\mathrm{degree}\left(\mathrm{u2}\,n\right),\mathrm{degree}\left(\mathrm{u3}\,n\right),\mathrm{degree}\left(\mathrm{u4}\,n\right)$

$19,0,7,3$

$\mathrm{degree}\left(\mathrm{u1}\,n\right)+\mathrm{degree}\left(\mathrm{v1}\,n\right),\mathrm{degree}\left(\mathrm{u2}\,n\right)+\mathrm{degree}\left(\mathrm{v2}\,n\right),\mathrm{degree}\left(\mathrm{u3}\,n\right)+\mathrm{degree}\left(\mathrm{v3}\,n\right),\mathrm{degree}\left(\mathrm{u4}\,n\right)+\mathrm{degree}\left(\mathrm{v4}\,n\right)$

$19,23,9,9$

Abramov, S.A.; Le, H.Q.; and Petkovsek, M. "Rational Canonical Forms and Efficient Representations of Hypergeometric Terms." Proc. ISSAC'2003, pp. 7-14. 2003.

Abramov, S.A., and Petkovsek, M. "Canonical representations of hypergeometric terms." Proc. FPSAC'2001, pp. 1-10. 2001.