For a specified bivariate hypergeometric term  in n and k, the ConjugateRTerm[1](T, n, k) and ConjugateRTerm[2](T, n, k) commands construct two r-terms conjugate to .

The output is a bivariate hypergeometric term, called an r-term, conjugate to , that is, it can be written as  where  is a rational function of n and k, and , a_i, b_i are integers, , , s, t are non-negative integers, and g_i, u, v are complex numbers.  is called a factorial term.

A polynomial  is integer-linear if it has the form  where a, b are integers, and c is a complex number.

For the first constructed r-term, all the integer-linear polynomials in the numerator and the denominator of the rational function  are moved into the factorial term .

For the second r-term, the integer-linear polynomials are moved from the factorial term  to the rational function , that is, for  such that , , then  is not an integer; and in the case that , either  or .

If the optional argument 'listform' is specified, the output is a list .

A sequence  is a bivariate hypergeometric term of n and k if there are nonzero polynomials , f_1, g_0, g_1 of n and k such that

for all non-negative integers n, k. Two hypergeometric terms T_1, T_2 are conjugate if they satisfy the above two relations with the same f_0, f_1, g_0, g_1.

Note: The ConjugateRTerm command replaces the CanonicalRepresentation command.

Abramov, S.A., and Petkovsek, M. "Canonical Representations of Hypergeometric Terms." Proceedings FPSAC'2001. pp. 1-10. 2001.

Abramov, S.A., and Petkovsek, M. "Proof of a Conjecture of Wilf and Zeilberger." University of Ljubljana, Preprint series. Vol. 39. (2001): 748.