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

The output is a bivariate hypergeometric term, called an r-term, conjugate to $T\left(n\,k\right)$, that is, it can be written as $R\left(n\,k\right)\mathrm{Tp}\left(n\,k\right)$ where $R\left(n\,k\right)$ is a rational function of n and k, and $\mathrm{Tp}\left(n\,k\right)=\frac{u^n{v}^k\left(\prod _{i=1}^s\left(b_{i}k+a_{i}n+g_i\right)!\right)}{\prod _{i=s+1}^t\left(a_i+b_i+g_i\right)!}$, a_i, b_i are integers, $\gcd \left(a_i\,b_i\right)=1$, $0\le a_i$, s, t are non-negative integers, and g_i, u, v are complex numbers. $\mathrm{Tp}\left(n\,k\right)$ is called a factorial term.

A polynomial $p\left(n\,k\right)$ is integer-linear if it has the form $an+bk+c$ 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 $R\left(n\,k\right)$ are moved into the factorial term $\mathrm{Tp}\left(n\,k\right)$.

For the second r-term, the integer-linear polynomials are moved from the factorial term $\mathrm{Tp}\left(n\,k\right)$ to the rational function $R\left(n\,k\right)$, that is, for $i\ne j$ such that $a_i=a_j$, $b_i=b_j$, then $g_i-g_j$ is not an integer; and in the case that $g_i-g_j=0$, either $i,j\le s$ or $i,s+1\le j$.

If the optional argument 'listform' is specified, the output is a list $\left[R\left(n\,k\right)\,\mathrm{Tp}\left(n\,k\right)\right]$.

A sequence $T\left(n\,k\right)$ is a bivariate hypergeometric term of n and k if there are nonzero polynomials $f_0$, 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.

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

$T≔\frac{2^{-k}\mathrm{binomial}\left(2n+k\,n\right)\left(-1+94n+83k\right)}{\left(n+k-2\right)\left(88n-53\right)}$

$T≔\frac{2^{-k}\left(\genfrac{}{}{0ex}{}{2n+k}{n}\right)\left(-1+94n+83k\right)}{\left(n+k-2\right)\left(88n-53\right)}$

$\mathrm{ConjugateRTerm}\left[1\right]\left(T\,n\,k\,'\mathrm{listform}'\right)$

$\left[-\frac{53}{7304}\,\frac{\left(-1+94n+83k\right)!\left(n+k-3\right)!{\left(\frac{1}{2}\right)}^k\left(n-\frac{141}{88}\right)!\left(2n+k\right)!}{\left(-2+94n+83k\right)!\left(n+k-2\right)!n!\left(n-\frac{53}{88}\right)!\left(n+k\right)!}\right]$

$\mathrm{ConjugateRTerm}\left[2\right]\left(T\,n\,k\,'\mathrm{listform}'\right)$

$\left[\frac{2809\left(-1+94n+83k\right)}{7304\left(88n-53\right)\left(n+k-2\right)}\,\frac{{\left(\frac{1}{2}\right)}^k\left(2n+k\right)!}{n!\left(n+k\right)!}\right]$

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.