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 th rational canonical form for F, .

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

The output is a sequence of 5 elements , called , where z is an element of K, and  are monic polynomials over K such that:

, .

 for all integers .

, .

, .

Note: Q is the automorphism of K(n) defined by .

The five-tuple  that satisfies the four conditions is a strict q-rational normal form for F. The rational function  and  are called the kernel and the shell of the , respectively.

Let  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  where p, q are polynomials in n, and G is a rational function of n, then  and .

Additionally, if  then  is minimal; if  then  is minimal; if  then  is minimal, and under this condition,  is minimal; if  then  is minimal, and under this condition,  is minimal.

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.