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QDifferenceEquations

 QMultiplicativeDecomposition
 construct the four minimal multiplicative decompositions of a q-hypergeometric term

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

Parameters

 H - q-hypergeometric term in q^n q - name used as the parameter q, usually q n - variable k - name

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{QDifferenceEquations}\right):$
 > $H≔\mathrm{Product}\left(\frac{\left({q}^{k}+{q}^{2}\right)\left({q}^{k}+1\right)\left({q}^{k}+{q}^{5}-{q}^{3}\right)\left({q}^{k}+{q}^{4}-{q}^{2}\right)\left({q}^{3}{q}^{k}+{q}^{2}-1\right)\left({q}^{12}{q}^{k}+{q}^{2}-1\right)}{\left({q}^{k}+{q}^{5}\right){\left({q}^{k}+{q}^{4}\right)}^{2}\left({q}^{4}{q}^{k}+1\right)\left({q}^{k}+{q}^{2}-1\right)\left({q}^{2}{q}^{k}+{q}^{2}-1\right)},k=0..n-1\right)$
 ${H}{≔}{\prod }_{{k}{=}{0}}^{{n}{-}{1}}{}\frac{\left({{q}}^{{k}}{+}{{q}}^{{2}}\right){}\left({{q}}^{{k}}{+}{1}\right){}\left({{q}}^{{k}}{+}{{q}}^{{5}}{-}{{q}}^{{3}}\right){}\left({{q}}^{{k}}{+}{{q}}^{{4}}{-}{{q}}^{{2}}\right){}\left({{q}}^{{3}}{}{{q}}^{{k}}{+}{{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{12}}{}{{q}}^{{k}}{+}{{q}}^{{2}}{-}{1}\right)}{\left({{q}}^{{k}}{+}{{q}}^{{5}}\right){}{\left({{q}}^{{k}}{+}{{q}}^{{4}}\right)}^{{2}}{}\left({{q}}^{{4}}{}{{q}}^{{k}}{+}{1}\right){}\left({{q}}^{{k}}{+}{{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{2}}{}{{q}}^{{k}}{+}{{q}}^{{2}}{-}{1}\right)}$ (1)
 > $\mathrm{QMultiplicativeDecomposition}\left[1\right]\left(H,q,n,k\right)$
 $\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}}}$ (2)
 > $\mathrm{QMultiplicativeDecomposition}\left[2\right]\left(H,q,n,k\right)$
 $\frac{{2}{}{\left({{q}}^{{3}}{-}{q}{+}{1}\right)}^{{2}}{}{\left({{q}}^{{4}}{-}{{q}}^{{2}}{+}{1}\right)}^{{2}}{}\left({1}{+}\frac{{1}}{{{q}}^{{3}}}\right){}\left({1}{+}\frac{{1}}{{{q}}^{{2}}}\right){}\left({1}{+}\frac{{1}}{{q}}\right){}\left({1}{+}\frac{{{q}}^{{2}}{-}{1}}{{q}}\right){}{{q}}^{{2}}{}\left({{q}}^{{5}}{-}{{q}}^{{3}}{+}{1}\right){}{\left({{q}}^{{18}}\right)}^{{n}}{}\left({{q}}^{{3}}{+}{{q}}^{{n}}\right){}\left({{q}}^{{4}}{+}{{q}}^{{n}}\right){}\left({\prod }_{{k}{=}{0}}^{{n}{-}{1}}{}\frac{\left({{q}}^{{k}}{+}\frac{{{q}}^{{2}}{-}{1}}{{{q}}^{{3}}}\right){}\left({{q}}^{{k}}{+}\frac{{{q}}^{{2}}{-}{1}}{{{q}}^{{12}}}\right)}{\left({{q}}^{{k}}{+}{{q}}^{{4}}\right){}\left({{q}}^{{k}}{+}{{q}}^{{5}}\right)}\right)}{\left({{q}}^{{3}}{+}{1}\right){}\left({{q}}^{{4}}{+}{1}\right){}{\left({{q}}^{{3}}{+}{{q}}^{{n}}{-}{q}\right)}^{{2}}{}{\left({{q}}^{{n}}{+}{{q}}^{{4}}{-}{{q}}^{{2}}\right)}^{{2}}{}\left({{q}}^{{n}}{+}\frac{{1}}{{{q}}^{{3}}}\right){}\left({{q}}^{{n}}{+}\frac{{1}}{{{q}}^{{2}}}\right){}\left({{q}}^{{n}}{+}\frac{{1}}{{q}}\right){}\left({{q}}^{{n}}{+}{1}\right){}\left({{q}}^{{n}}{+}\frac{{{q}}^{{2}}{-}{1}}{{q}}\right){}\left({{q}}^{{2}}{+}{{q}}^{{n}}{-}{1}\right){}\left({{q}}^{{n}}{+}{{q}}^{{5}}{-}{{q}}^{{3}}\right)}$ (3)
 > $\mathrm{QMultiplicativeDecomposition}\left[3\right]\left(H,q,n,k\right)$
 $\frac{\left({{q}}^{{3}}{-}{q}{+}{1}\right){}\left({{q}}^{{4}}{-}{{q}}^{{2}}{+}{1}\right){}{\left({{q}}^{{4}}\right)}^{{n}}{}{\left({{q}}^{{3}}{+}{{q}}^{{n}}\right)}^{{2}}{}{\left({{q}}^{{4}}{+}{{q}}^{{n}}\right)}^{{2}}{}\left({q}{+}{{q}}^{{n}}\right){}\left({{q}}^{{2}}{+}{{q}}^{{n}}\right){}\left({{q}}^{{n}}{+}\frac{{{q}}^{{2}}{-}{1}}{{{q}}^{{2}}}\right){}\left({\prod }_{{k}{=}{0}}^{{n}{-}{1}}{}\frac{\left({{q}}^{{k}}{+}{{q}}^{{5}}{-}{{q}}^{{3}}\right){}\left({{q}}^{{k}}{+}\frac{{{q}}^{{2}}{-}{1}}{{{q}}^{{12}}}\right)}{\left({{q}}^{{k}}{+}{{q}}^{{5}}\right){}\left({{q}}^{{k}}{+}\frac{{1}}{{{q}}^{{4}}}\right)}\right)}{{\left({{q}}^{{3}}{+}{1}\right)}^{{2}}{}{\left({{q}}^{{4}}{+}{1}\right)}^{{2}}{}\left({q}{+}{1}\right){}\left({{q}}^{{2}}{+}{1}\right){}\left({1}{+}\frac{{{q}}^{{2}}{-}{1}}{{{q}}^{{2}}}\right){}\left({{q}}^{{3}}{+}{{q}}^{{n}}{-}{q}\right){}\left({{q}}^{{n}}{+}{{q}}^{{4}}{-}{{q}}^{{2}}\right)}$ (4)
 > $\mathrm{QMultiplicativeDecomposition}\left[4\right]\left(H,q,n,k\right)$
 $\frac{{2}{}\left({{q}}^{{3}}{-}{q}{+}{1}\right){}\left({{q}}^{{4}}{-}{{q}}^{{2}}{+}{1}\right){}\left({1}{+}\frac{{1}}{{{q}}^{{3}}}\right){}\left({1}{+}\frac{{1}}{{{q}}^{{2}}}\right){}\left({1}{+}\frac{{1}}{{q}}\right){}{\left({{q}}^{{12}}\right)}^{{n}}{}\left({{q}}^{{n}}{+}\frac{{{q}}^{{2}}{-}{1}}{{{q}}^{{2}}}\right){}\left({{q}}^{{3}}{+}{{q}}^{{n}}\right){}\left({{q}}^{{4}}{+}{{q}}^{{n}}\right){}\left({\prod }_{{k}{=}{0}}^{{n}{-}{1}}{}\frac{\left({{q}}^{{k}}{+}{{q}}^{{5}}{-}{{q}}^{{3}}\right){}\left({{q}}^{{k}}{+}\frac{{{q}}^{{2}}{-}{1}}{{{q}}^{{12}}}\right)}{\left({{q}}^{{k}}{+}{{q}}^{{5}}\right){}\left({{q}}^{{k}}{+}{{q}}^{{4}}\right)}\right)}{\left({1}{+}\frac{{{q}}^{{2}}{-}{1}}{{{q}}^{{2}}}\right){}\left({{q}}^{{3}}{+}{1}\right){}\left({{q}}^{{4}}{+}{1}\right){}\left({{q}}^{{3}}{+}{{q}}^{{n}}{-}{q}\right){}\left({{q}}^{{n}}{+}{{q}}^{{4}}{-}{{q}}^{{2}}\right){}\left({{q}}^{{n}}{+}\frac{{1}}{{{q}}^{{3}}}\right){}\left({{q}}^{{n}}{+}\frac{{1}}{{{q}}^{{2}}}\right){}\left({{q}}^{{n}}{+}\frac{{1}}{{q}}\right){}\left({{q}}^{{n}}{+}{1}\right)}$ (5)

References

 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.