MultiplicativeDecomposition - Maple Help

SumTools[Hypergeometric]

 MultiplicativeDecomposition
 construct the four minimal multiplicative decompositions of a hypergeometric term

 Calling Sequence MultiplicativeDecomposition[1](H, n, k) MultiplicativeDecomposition[2](H, n, k) MultiplicativeDecomposition[3](H, n, k) MultiplicativeDecomposition[4](H, n, k)

Parameters

 H - hypergeometric term of n n - variable k - name

Description

 • Let H be a hypergeometric term of n. The MultiplicativeDecomposition[i](H,n,k) calling sequence constructs the ith minimal multiplicative decomposition of H of the form $H\left(n\right)=W\left(n\right)\left({\prod }_{k=\mathrm{n0}}^{n-1}F\left(k\right)\right)$ where $W\left(n\right),F\left(n\right)$ are rational functions of n, $\mathrm{degree}\left(\mathrm{numer}\left(F\left(n\right)\right)\right)$ and $\mathrm{degree}\left(\mathrm{denom}\left(F\left(n\right)\right)\right)$ have minimal possible values, for $i=\left\{1,2,3,4\right\}$.
 If $i=1$ then $\mathrm{degree}\left(\mathrm{denom}\left(W\right)\right)$ is minimal.
 If $i=2$ then $\mathrm{degree}\left(\mathrm{numer}\left(W\right)\right)$ is minimal.
 If $i=3$ then $\mathrm{degree}\left(\mathrm{numer}\left(W\right)\right)+\mathrm{degree}\left(\mathrm{denom}\left(W\right)\right)$ is minimal, and $\mathrm{degree}\left(\mathrm{denom}\left(W\right)\right)$ is minimal.
 If $i=4$ then $\mathrm{degree}\left(\mathrm{numer}\left(W\right)\right)+\mathrm{degree}\left(\mathrm{denom}\left(W\right)\right)$ is minimal, and $\mathrm{degree}\left(\mathrm{numer}\left(W\right)\right)$ is minimal.
 If the MultiplicativeDecomposition command is called without an index, the first minimal multiplicative decomposition is constructed.

Examples

 > $\mathrm{with}\left(\mathrm{SumTools}\left[\mathrm{Hypergeometric}\right]\right):$
 > $H≔24\left(\mathrm{Product}\left(\frac{k\left(k+2\right)\left(k-4+{2}^{\frac{1}{2}}\right)\left(k-3+{2}^{\frac{1}{2}}\right)\left(k+2+{2}^{\frac{1}{2}}\right)\left(k+11+{2}^{\frac{1}{2}}\right)}{\left(k-3\right){\left(k-2\right)}^{2}\left(k+6\right)\left(k+12\right)\left(k-1+{2}^{\frac{1}{2}}\right)\left(k+1+{2}^{\frac{1}{2}}\right)},k=4..n-1\right)\right)$
 ${H}{≔}{24}{}\left({\prod }_{{k}{=}{4}}^{{n}{-}{1}}{}\frac{{k}{}\left({k}{+}{2}\right){}\left({k}{-}{4}{+}\sqrt{{2}}\right){}\left({k}{-}{3}{+}\sqrt{{2}}\right){}\left({k}{+}{2}{+}\sqrt{{2}}\right){}\left({k}{+}{11}{+}\sqrt{{2}}\right)}{\left({k}{-}{3}\right){}{\left({k}{-}{2}\right)}^{{2}}{}\left({k}{+}{6}\right){}\left({k}{+}{12}\right){}\left({k}{-}{1}{+}\sqrt{{2}}\right){}\left({k}{+}{1}{+}\sqrt{{2}}\right)}\right)$ (1)
 > $\mathrm{MultiplicativeDecomposition}\left[1\right]\left(H,n,k\right)$
 $\frac{{\left({n}{+}{1}{+}\sqrt{{2}}\right)}^{{2}}{}{\left({n}{-}{1}\right)}^{{2}}{}{\left({n}{-}{2}\right)}^{{2}}{}\left({n}{+}{1}\right){}{n}{}\left({n}{+}{10}{+}\sqrt{{2}}\right){}\left({n}{+}{9}{+}\sqrt{{2}}\right){}\left({n}{+}{8}{+}\sqrt{{2}}\right){}\left({n}{+}{7}{+}\sqrt{{2}}\right){}\left({n}{+}{6}{+}\sqrt{{2}}\right){}\left({n}{+}{5}{+}\sqrt{{2}}\right){}\left({n}{+}{4}{+}\sqrt{{2}}\right){}\left({n}{+}{3}{+}\sqrt{{2}}\right){}\left({n}{+}{2}{+}\sqrt{{2}}\right){}\left({n}{+}\sqrt{{2}}\right){}\left({n}{-}{1}{+}\sqrt{{2}}\right){}\left({\prod }_{{k}{=}{4}}^{{n}{-}{1}}{}\frac{\left({k}{-}{4}{+}\sqrt{{2}}\right){}\left({k}{-}{3}{+}\sqrt{{2}}\right)}{\left({k}{-}{3}\right){}\left({k}{+}{6}\right){}\left({k}{+}{12}\right)}\right)}{{30}{}{\left({5}{+}\sqrt{{2}}\right)}^{{2}}{}\left({14}{+}\sqrt{{2}}\right){}\left({13}{+}\sqrt{{2}}\right){}\left({12}{+}\sqrt{{2}}\right){}\left({11}{+}\sqrt{{2}}\right){}\left({10}{+}\sqrt{{2}}\right){}\left({9}{+}\sqrt{{2}}\right){}\left({8}{+}\sqrt{{2}}\right){}\left({7}{+}\sqrt{{2}}\right){}\left({6}{+}\sqrt{{2}}\right){}\left({4}{+}\sqrt{{2}}\right){}\left({3}{+}\sqrt{{2}}\right)}$ (2)
 > $\mathrm{MultiplicativeDecomposition}\left[2\right]\left(H,n,k\right)$
 $\frac{{15817629155328000}{}{\left({2}{+}\sqrt{{2}}\right)}^{{2}}{}{\left({1}{+}\sqrt{{2}}\right)}^{{2}}{}\left({4}{+}\sqrt{{2}}\right){}\left({3}{+}\sqrt{{2}}\right){}\sqrt{{2}}{}\left({\prod }_{{k}{=}{4}}^{{n}{-}{1}}{}\frac{\left({k}{+}{2}{+}\sqrt{{2}}\right){}\left({k}{+}{11}{+}\sqrt{{2}}\right)}{{\left({k}{-}{2}\right)}^{{2}}{}\left({k}{-}{3}\right)}\right)}{{\left({n}{-}{2}{+}\sqrt{{2}}\right)}^{{2}}{}{\left({n}{-}{3}{+}\sqrt{{2}}\right)}^{{2}}{}{\left({n}{+}{5}\right)}^{{2}}{}{\left({n}{+}{4}\right)}^{{2}}{}{\left({n}{+}{3}\right)}^{{2}}{}{\left({n}{+}{2}\right)}^{{2}}{}\left({n}{+}\sqrt{{2}}\right){}\left({n}{-}{1}{+}\sqrt{{2}}\right){}\left({n}{-}{4}{+}\sqrt{{2}}\right){}\left({n}{+}{11}\right){}\left({n}{+}{10}\right){}\left({n}{+}{9}\right){}\left({n}{+}{8}\right){}\left({n}{+}{7}\right){}\left({n}{+}{6}\right){}\left({n}{+}{1}\right){}{n}}$ (3)
 > $\mathrm{MultiplicativeDecomposition}\left[3\right]\left(H,n,k\right)$
 $\frac{\left({2}{+}\sqrt{{2}}\right){}\left({1}{+}\sqrt{{2}}\right){}\left({n}{+}{1}{+}\sqrt{{2}}\right){}{\left({n}{-}{1}\right)}^{{2}}{}{\left({n}{-}{2}\right)}^{{2}}{}\left({n}{+}{1}\right){}{n}{}\left({\prod }_{{k}{=}{4}}^{{n}{-}{1}}{}\frac{\left({k}{-}{4}{+}\sqrt{{2}}\right){}\left({k}{+}{11}{+}\sqrt{{2}}\right)}{\left({k}{-}{3}\right){}\left({k}{+}{6}\right){}\left({k}{+}{12}\right)}\right)}{{30}{}\left({5}{+}\sqrt{{2}}\right){}\left({n}{-}{2}{+}\sqrt{{2}}\right){}\left({n}{-}{3}{+}\sqrt{{2}}\right)}$ (4)
 > $\mathrm{MultiplicativeDecomposition}\left[4\right]\left(H,n,k\right)$
 $\frac{{12096}{}\left({2}{+}\sqrt{{2}}\right){}\left({1}{+}\sqrt{{2}}\right){}\left({n}{+}{1}{+}\sqrt{{2}}\right){}\left({n}{-}{1}\right){}\left({n}{-}{2}\right){}\left({\prod }_{{k}{=}{4}}^{{n}{-}{1}}{}\frac{\left({k}{-}{4}{+}\sqrt{{2}}\right){}\left({k}{+}{11}{+}\sqrt{{2}}\right)}{\left({k}{-}{3}\right){}\left({k}{-}{2}\right){}\left({k}{+}{12}\right)}\right)}{\left({5}{+}\sqrt{{2}}\right){}\left({n}{-}{2}{+}\sqrt{{2}}\right){}\left({n}{-}{3}{+}\sqrt{{2}}\right){}\left({n}{+}{5}\right){}\left({n}{+}{4}\right){}\left({n}{+}{3}\right){}\left({n}{+}{2}\right)}$ (5)

References

 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.