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, .

If the RationalCanonicalForm command is called without an index, the first 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 k.

, .

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

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

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

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. "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.