Let H be a q-hypergeometric term in q^n. The QMultiplicativeDecomposition[i](H,q,n,k) command constructs the $i$th minimal multiplicative decomposition of H of the form $H\left(q^n\right)=W\left(q^n\right)\left(\prod _{k=\mathrm{n0}}^{n-1}F\left(q^k\right)\right)$ where $W\left(q^n\right),F\left(q^n\right)$ are rational functions of q^n, $\mathrm{degree}\left(\mathrm{numer}\left(F\left(q^n\right)\right)\right)$ and $\mathrm{degree}\left(\mathrm{denom}\left(F\left(q^n\right)\right)\right)$ have minimal possible values, for $i=\left\{1\,2\,3\,4\right\}$.

Additionally, 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 under this condition, $\mathrm{degree}\left(\mathrm{denom}\left(W\right)\right)$ 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 under this condition, $\mathrm{degree}\left(\mathrm{numer}\left(W\right)\right)$ is minimal.

If QMultiplicativeDecomposition is called without an index, the first minimal multiplicative decomposition is constructed.

$\mathrm{with}\left(\mathrm{QDifferenceEquations}\right)\:$

$H≔\mathrm{Product}\left(\frac{\left(q^k+q^2\right)\left(q^k+1\right)\left(q^k+q^5-q^3\right)\left(q^k+q^4-q^2\right)\left(q^3{q}^k+q^2-1\right)\left(q^{12}{q}^k+q^2-1\right)}{\left(q^k+q^5\right){\left(q^k+q^4\right)}^2\left(q^4{q}^k+1\right)\left(q^k+q^2-1\right)\left(q^2{q}^k+q^2-1\right)}\,k=0..n-1\right)$

$H≔\prod _{k=0}^{n-1}\frac{\left(q^k+q^2\right)\left(q^k+1\right)\left(q^k+q^5-q^3\right)\left(q^k+q^4-q^2\right)\left(q^3{q}^k+q^2-1\right)\left(q^{12}{q}^k+q^2-1\right)}{\left(q^k+q^5\right){\left(q^k+q^4\right)}^2\left(q^4{q}^k+1\right)\left(q^k+q^2-1\right)\left(q^2{q}^k+q^2-1\right)}$

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

$\frac{{\left(\frac{1}{q^{10}}\right)}^n{\left(q^n+\frac{q^2-1}{q^2}\right)}^2{\left(q^3+q^n\right)}^2{\left(q^4+q^n\right)}^2\left(q+q^n\right)\left(q^2+q^n\right)\left(q^n+\frac{q^2-1}{q^{11}}\right)\left(q^n+\frac{q^2-1}{q^{10}}\right)\left(q^n+\frac{q^2-1}{q^9}\right)\left(q^n+\frac{q^2-1}{q^8}\right)\left(q^n+\frac{q^2-1}{q^7}\right)\left(q^n+\frac{q^2-1}{q^6}\right)\left(q^n+\frac{q^2-1}{q^5}\right)\left(q^n+\frac{q^2-1}{q^4}\right)\left(q^n+\frac{q^2-1}{q^3}\right)\left(q^n+\frac{q^2-1}{q}\right)\left(q^2+q^n-1\right)\left(\prod _{k=0}^{n-1}\frac{\left(q^k+q^5-q^3\right)\left(q^k+q^4-q^2\right)}{\left(q^k+q^5\right)\left(q^k+\frac{1}{q^4}\right)}\right)}{{\left(1+\frac{q^2-1}{q^2}\right)}^2{\left(q^3+1\right)}^2{\left(q^4+1\right)}^2\left(q+1\right)\left(q^2+1\right)\left(1+\frac{q^2-1}{q^{11}}\right)\left(1+\frac{q^2-1}{q^{10}}\right)\left(1+\frac{q^2-1}{q^9}\right)\left(1+\frac{q^2-1}{q^8}\right)\left(1+\frac{q^2-1}{q^7}\right)\left(1+\frac{q^2-1}{q^6}\right)\left(1+\frac{q^2-1}{q^5}\right)\left(1+\frac{q^2-1}{q^4}\right)\left(1+\frac{q^2-1}{q^3}\right)\left(1+\frac{q^2-1}{q}\right)q^2}$

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

$\frac{2{\left(q^3-q+1\right)}^2{\left(q^4-q^2+1\right)}^2\left(1+\frac{1}{q^3}\right)\left(1+\frac{1}{q^2}\right)\left(1+\frac{1}{q}\right)\left(1+\frac{q^2-1}{q}\right)q^2\left(q^5-q^3+1\right){\left(q^{18}\right)}^n\left(q^3+q^n\right)\left(q^4+q^n\right)\left(\prod _{k=0}^{n-1}\frac{\left(q^k+\frac{q^2-1}{q^3}\right)\left(q^k+\frac{q^2-1}{q^{12}}\right)}{\left(q^k+q^4\right)\left(q^k+q^5\right)}\right)}{\left(q^3+1\right)\left(q^4+1\right){\left(q^3+q^n-q\right)}^2{\left(q^n+q^4-q^2\right)}^2\left(q^n+\frac{1}{q^3}\right)\left(q^n+\frac{1}{q^2}\right)\left(q^n+\frac{1}{q}\right)\left(q^n+1\right)\left(q^n+\frac{q^2-1}{q}\right)\left(q^2+q^n-1\right)\left(q^n+q^5-q^3\right)}$

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

$\frac{\left(q^3-q+1\right)\left(q^4-q^2+1\right){\left(q^4\right)}^n{\left(q^3+q^n\right)}^2{\left(q^4+q^n\right)}^2\left(q+q^n\right)\left(q^2+q^n\right)\left(q^n+\frac{q^2-1}{q^2}\right)\left(\prod _{k=0}^{n-1}\frac{\left(q^k+q^5-q^3\right)\left(q^k+\frac{q^2-1}{q^{12}}\right)}{\left(q^k+q^5\right)\left(q^k+\frac{1}{q^4}\right)}\right)}{{\left(q^3+1\right)}^2{\left(q^4+1\right)}^2\left(q+1\right)\left(q^2+1\right)\left(1+\frac{q^2-1}{q^2}\right)\left(q^3+q^n-q\right)\left(q^n+q^4-q^2\right)}$

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

$\frac{2\left(1+\frac{1}{q^3}\right)\left(1+\frac{1}{q^2}\right)\left(1+\frac{1}{q}\right)\left(q^3-q+1\right)\left(q^4-q^2+1\right){\left(q^{12}\right)}^n\left(q^3+q^n\right)\left(q^4+q^n\right)\left(q^n+\frac{q^2-1}{q^2}\right)\left(\prod _{k=0}^{n-1}\frac{\left(q^k+q^5-q^3\right)\left(q^k+\frac{q^2-1}{q^{12}}\right)}{\left(q^k+q^5\right)\left(q^k+q^4\right)}\right)}{\left(q^3+1\right)\left(q^4+1\right)\left(1+\frac{q^2-1}{q^2}\right)\left(q^n+\frac{1}{q^3}\right)\left(q^n+\frac{1}{q^2}\right)\left(q^n+\frac{1}{q}\right)\left(q^n+1\right)\left(q^3+q^n-q\right)\left(q^n+q^4-q^2\right)}$

Abramov, S.A.; Le, H.Q.; and Petkovsek, M. "Efficient Representations of (q-)Hypergeometric Terms and the Assignment Problem." Submitted.

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.