Let F be a rational function of n over a field K of characteristic 0, q is a nonzero element of K which is not a root of unity. The QRationalCanonicalForm[i](F,q,n) command constructs the $i$th rational canonical form for F, $i=\left\{1\,2\,3\,4\right\}$.

If QRationalCanonicalForm is called without an index, the first q-rational canonical form is constructed.

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

$F=\frac{z\left(\frac{r}{s}\right)Q\left(\frac{u}{v}\right)}{\left(\frac{u}{v}\right)}$, $\gcd \left(u\,v\right)=1$.

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

$u\left(0\right)\ne 0$, $v\left(0\right)\ne 0$.

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

Note: Q is the automorphism of K(n) defined by $Q\left(F\left(n\right)\right)=F\left(qn\right)$.

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

Let $\mathrm{\phi }=\left(z\,r\,s\,u\,v\right)$ be any qRNF 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 $F\left(n\right)=\frac{p\left(n\right)Q\left(G\left(n\right)\right)}{q\left(n\right)G\left(n\right)}$ 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 $\mathrm{degree}\left(s\right)\le \mathrm{degree}\left(q\right)$.

Additionally, 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.

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

$\mathrm{\nu }≔\left(n+q^2\right)q^{11}\left(n+1\right)\left(n+q^5-q^3\right)\left(n+q^4-q^2\right)\left(q^{3}n+q^2-1\right)\left(q^{12}n+q^2-1\right)\:$

$\mathrm{de}≔\left(n+q^5\right){\left(n+q^4\right)}^2{q}^{11}\left(q^{4}n+1\right)\left(n+q^2-1\right)\left(q^{2}n+q^2-1\right)\:$

$F≔\frac{\mathrm{\nu }}{\mathrm{de}}$

$F≔\frac{\left(q^2+n\right)\left(n+1\right)\left(q^5-q^3+n\right)\left(q^4-q^2+n\right)\left(q^{3}n+q^2-1\right)\left(q^{12}n+q^2-1\right)}{\left(q^5+n\right){\left(q^4+n\right)}^2\left(q^{4}n+1\right)\left(q^2+n-1\right)\left(q^{2}n+q^2-1\right)}$

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

$\mathrm{z1},\mathrm{r1},\mathrm{s1},\mathrm{u1},\mathrm{v1}≔\frac{1}{q^{10}},\left(q^5-q^3+n\right)\left(q^4-q^2+n\right),\left(q^5+n\right)\left(n+\frac{1}{q^4}\right),{\left(n+\frac{q^2-1}{q^2}\right)}^2{\left(q^3+n\right)}^2{\left(q^4+n\right)}^2\left(q+n\right)\left(q^2+n\right)\left(n+\frac{q^2-1}{q^{11}}\right)\left(n+\frac{q^2-1}{q^{10}}\right)\left(n+\frac{q^2-1}{q^9}\right)\left(n+\frac{q^2-1}{q^8}\right)\left(n+\frac{q^2-1}{q^7}\right)\left(n+\frac{q^2-1}{q^6}\right)\left(n+\frac{q^2-1}{q^5}\right)\left(n+\frac{q^2-1}{q^4}\right)\left(n+\frac{q^2-1}{q^3}\right)\left(n+\frac{q^2-1}{q}\right)\left(q^2+n-1\right),1$

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

$\mathrm{z2},\mathrm{r2},\mathrm{s2},\mathrm{u2},\mathrm{v2}≔q^{18},\left(n+\frac{q^2-1}{q^3}\right)\left(n+\frac{q^2-1}{q^{12}}\right),\left(q^4+n\right)\left(q^5+n\right),\left(q^3+n\right)\left(q^4+n\right),{\left(q^3+n-q\right)}^2{\left(q^4-q^2+n\right)}^2\left(n+\frac{1}{q^3}\right)\left(n+\frac{1}{q^2}\right)\left(n+\frac{1}{q}\right)\left(n+1\right)\left(n+\frac{q^2-1}{q}\right)\left(q^2+n-1\right)\left(q^5-q^3+n\right)$

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

$\mathrm{z3},\mathrm{r3},\mathrm{s3},\mathrm{u3},\mathrm{v3}≔q^4,\left(q^5-q^3+n\right)\left(n+\frac{q^2-1}{q^{12}}\right),\left(q^5+n\right)\left(n+\frac{1}{q^4}\right),{\left(q^3+n\right)}^2{\left(q^4+n\right)}^2\left(q+n\right)\left(q^2+n\right)\left(n+\frac{q^2-1}{q^2}\right),\left(q^3+n-q\right)\left(q^4-q^2+n\right)$

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

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

$\mathrm{normal}\left(F-\frac{\mathrm{z2}\left(\frac{\mathrm{r2}}{\mathrm{s2}}\right)\mathrm{subs}\left(n=qn\,\frac{\mathrm{u2}}{\mathrm{v2}}\right)}{\frac{\mathrm{u2}}{\mathrm{v2}}}\right),\mathrm{gcdex}\left(\mathrm{u2}\,\mathrm{v2}\,n\right)$

$0,1$

$\mathrm{QDispersion}\left(\mathrm{r2}\,\mathrm{s2}\,q\,n\right),\mathrm{QDispersion}\left(\mathrm{s2}\,\mathrm{r2}\,q\,n\right)$

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

$\mathrm{eval}\left(\mathrm{u2}\,n=0\right)\ne 0,\mathrm{normal}\left(\mathrm{eval}\left(\mathrm{v2}\,n=0\right)\right)\ne 0$

$q^7\ne 0,{\left(q^2-1\right)}^7{q}^2\ne 0$

$\mathrm{gcdex}\left(\mathrm{r2}\,\mathrm{u2}\mathrm{subs}\left(n=qn\,\mathrm{v2}\right)\,n\right),\mathrm{gcdex}\left(\mathrm{s2}\,\mathrm{subs}\left(n=qn\,\mathrm{u2}\right)\mathrm{v2}\,n\right)$

$1,1$

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

$2,2,2,2$

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

$2,2,2,2$

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

$0,11,2,6$

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

$19,2,7,3$

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

$19,13,9,9$

Abramov, S.A.; Le, H.Q.; and Petkovsek, M. "Efficient Representations of (q-)Hypergeometric Terms and the Assignment Problem." Submitted.

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.