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.

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

$H≔24\left(\mathrm{Product}\left(\frac{k\left(k+2\right)\left(k-4+2^{\frac{1}{2}}\right)\left(k-3+2^{\frac{1}{2}}\right)\left(k+2+2^{\frac{1}{2}}\right)\left(k+11+2^{\frac{1}{2}}\right)}{\left(k-3\right){\left(k-2\right)}^2\left(k+6\right)\left(k+12\right)\left(k-1+2^{\frac{1}{2}}\right)\left(k+1+2^{\frac{1}{2}}\right)}\,k=4..n-1\right)\right)$

$H≔24\left(\prod _{k=4}^{n-1}\frac{k\left(k+2\right)\left(k-4+\sqrt{2}\right)\left(k-3+\sqrt{2}\right)\left(k+2+\sqrt{2}\right)\left(k+11+\sqrt{2}\right)}{\left(k-3\right){\left(k-2\right)}^2\left(k+6\right)\left(k+12\right)\left(k-1+\sqrt{2}\right)\left(k+1+\sqrt{2}\right)}\right)$

$\mathrm{MultiplicativeDecomposition}\left[1\right]\left(H\,n\,k\right)$

$\frac{{\left(n+1+\sqrt{2}\right)}^2{\left(n-1\right)}^2{\left(n-2\right)}^2\left(n+1\right)n\left(n+10+\sqrt{2}\right)\left(n+9+\sqrt{2}\right)\left(n+8+\sqrt{2}\right)\left(n+7+\sqrt{2}\right)\left(n+6+\sqrt{2}\right)\left(n+5+\sqrt{2}\right)\left(n+4+\sqrt{2}\right)\left(n+3+\sqrt{2}\right)\left(n+2+\sqrt{2}\right)\left(n+\sqrt{2}\right)\left(n-1+\sqrt{2}\right)\left(\prod _{k=4}^{n-1}\frac{\left(k-4+\sqrt{2}\right)\left(k-3+\sqrt{2}\right)}{\left(k+12\right)\left(k+6\right)\left(k-3\right)}\right)}{30{\left(5+\sqrt{2}\right)}^2\left(14+\sqrt{2}\right)\left(13+\sqrt{2}\right)\left(12+\sqrt{2}\right)\left(11+\sqrt{2}\right)\left(10+\sqrt{2}\right)\left(9+\sqrt{2}\right)\left(8+\sqrt{2}\right)\left(7+\sqrt{2}\right)\left(6+\sqrt{2}\right)\left(4+\sqrt{2}\right)\left(3+\sqrt{2}\right)}$

$\mathrm{MultiplicativeDecomposition}\left[2\right]\left(H\,n\,k\right)$

$\frac{15817629155328000{\left(2+\sqrt{2}\right)}^2{\left(1+\sqrt{2}\right)}^2\left(4+\sqrt{2}\right)\left(3+\sqrt{2}\right)\sqrt{2}\left(\prod _{k=4}^{n-1}\frac{\left(k+2+\sqrt{2}\right)\left(k+11+\sqrt{2}\right)}{\left(k-3\right){\left(k-2\right)}^2}\right)}{{\left(n-2+\sqrt{2}\right)}^2{\left(n-3+\sqrt{2}\right)}^2{\left(n+5\right)}^2{\left(n+4\right)}^2{\left(n+3\right)}^2{\left(n+2\right)}^2\left(n+\sqrt{2}\right)\left(n-1+\sqrt{2}\right)\left(n-4+\sqrt{2}\right)\left(n+11\right)\left(n+10\right)\left(n+9\right)\left(n+8\right)\left(n+7\right)\left(n+6\right)\left(n+1\right)n}$

$\mathrm{MultiplicativeDecomposition}\left[3\right]\left(H\,n\,k\right)$

$\frac{\left(2+\sqrt{2}\right)\left(1+\sqrt{2}\right)\left(n+1+\sqrt{2}\right){\left(n-1\right)}^2{\left(n-2\right)}^2\left(n+1\right)n\left(\prod _{k=4}^{n-1}\frac{\left(k-4+\sqrt{2}\right)\left(k+11+\sqrt{2}\right)}{\left(k+12\right)\left(k+6\right)\left(k-3\right)}\right)}{30\left(5+\sqrt{2}\right)\left(n-2+\sqrt{2}\right)\left(n-3+\sqrt{2}\right)}$

$\mathrm{MultiplicativeDecomposition}\left[4\right]\left(H\,n\,k\right)$

$\frac{12096\left(2+\sqrt{2}\right)\left(1+\sqrt{2}\right)\left(n+1+\sqrt{2}\right)\left(n-1\right)\left(n-2\right)\left(\prod _{k=4}^{n-1}\frac{\left(k-4+\sqrt{2}\right)\left(k+11+\sqrt{2}\right)}{\left(k-3\right)\left(k-2\right)\left(k+12\right)}\right)}{\left(5+\sqrt{2}\right)\left(n-2+\sqrt{2}\right)\left(n-3+\sqrt{2}\right)\left(n+5\right)\left(n+4\right)\left(n+3\right)\left(n+2\right)}$

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.