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SumTools[Hypergeometric]

 ConjugateRTerm
 construct r-terms conjugate to a bivariate hypergeometric term

 Calling Sequence ConjugateRTerm[1](T, n, k, 'listform') ConjugateRTerm[2](T, n, k, 'listform')

Parameters

 T - hypergeometric term of n and k n - name k - name 'listform' - (optional) specify output as a list

Description

 • 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, $\mathrm{gcd}\left({a}_{i},{b}_{i}\right)=1$, $0\le {a}_{i}$, 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

${f}_{1}\left(n,k\right)T\left(n+1,k\right)={f}_{0}\left(n,k\right)T\left(n,k\right),{g}_{1}\left(n,k\right)T\left(n,k+1\right)={g}_{0}\left(n,k\right)T\left(n,k\right)$

 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.

Examples

 > $\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{Hypergeometric}\right]\right):$
 > $T≔\frac{{2}^{-k}\mathrm{binomial}\left(2n+k,n\right)\left(-1+94n+83k\right)}{\left(n+k-2\right)\left(88n-53\right)}$
 ${T}{≔}\frac{{{2}}^{{-}{k}}{}\left(\genfrac{}{}{0}{}{{2}{}{n}{+}{k}}{{n}}\right){}\left({-}{1}{+}{94}{}{n}{+}{83}{}{k}\right)}{\left({n}{+}{k}{-}{2}\right){}\left({88}{}{n}{-}{53}\right)}$ (1)
 > $\mathrm{ConjugateRTerm}\left[1\right]\left(T,n,k,'\mathrm{listform}'\right)$
 $\left[{-}\frac{{53}}{{7304}}{,}\frac{\left({-}{1}{+}{94}{}{n}{+}{83}{}{k}\right){!}{}\left({n}{+}{k}{-}{3}\right){!}{}{\left(\frac{{1}}{{2}}\right)}^{{k}}{}\left({n}{-}\frac{{141}}{{88}}\right){!}{}\left({2}{}{n}{+}{k}\right){!}}{\left({-}{2}{+}{94}{}{n}{+}{83}{}{k}\right){!}{}\left({n}{+}{k}{-}{2}\right){!}{}{n}{!}{}\left({n}{-}\frac{{53}}{{88}}\right){!}{}\left({n}{+}{k}\right){!}}\right]$ (2)
 > $\mathrm{ConjugateRTerm}\left[2\right]\left(T,n,k,'\mathrm{listform}'\right)$
 $\left[\frac{{2809}{}\left({-}{1}{+}{94}{}{n}{+}{83}{}{k}\right)}{{7304}{}\left({88}{}{n}{-}{53}\right){}\left({n}{+}{k}{-}{2}\right)}{,}\frac{{\left(\frac{{1}}{{2}}\right)}^{{k}}{}\left({2}{}{n}{+}{k}\right){!}}{{n}{!}{}\left({n}{+}{k}\right){!}}\right]$ (3)

References

 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.