QEfficientRepresentation - Maple Help

QDifferenceEquations

 QEfficientRepresentation
 construct the four efficient representations of a q-hypergeometric term

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

Parameters

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

Description

 • Let H be a q-hypergeometric term in ${q}^{n}$. The QEfficientRepresentation[i](H,q,n) command constructs the ith efficient representation of H of the form $H\left(n\right)=C{\mathrm{\alpha }}^{n}V\left({q}^{n}\right)Q\left(n\right)$ where $C$, $\mathrm{\alpha }$ are constant and $Q\left(n\right)$ is a product of QPochhammer-function values and their reciprocals. Additionally,
 1 $Q\left(n\right)$ has the minimal number of factors,
 2 $V\left({q}^{n}\right)$ is a rational function which is minimal in one sense or another, depending on the particular q-rational canonical form chosen to represent the certificate of $H\left({q}^{n}\right)$.
 • If $i=1$ then $\mathrm{degree}\left(\mathrm{denom}\left(V\right)\right)$ is minimal; if $i=2$ then $\mathrm{degree}\left(\mathrm{numer}\left(V\right)\right)$ is minimal; if $i=3$ then $\mathrm{degree}\left(\mathrm{numer}\left(V\right)\right)+\mathrm{degree}\left(\mathrm{denom}\left(V\right)\right)$ is minimal, and under this condition, $\mathrm{degree}\left(\mathrm{denom}\left(V\right)\right)$ is minimal; if $i=4$ then $\mathrm{degree}\left(\mathrm{numer}\left(V\right)\right)+\mathrm{degree}\left(\mathrm{denom}\left(V\right)\right)$ is minimal, and under this condition, $\mathrm{degree}\left(\mathrm{numer}\left(V\right)\right)$ is minimal.
 • If QEfficientRepresentation is called without an index, the first efficient representation 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)
 > $\mathrm{QEfficientRepresentation}\left[1\right]\left(H,q,n\right)$
 $\frac{{{q}}^{{66}}{}{\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(\frac{{\left({{q}}^{{2}}{-}{1}\right)}^{{2}}}{{{q}}^{{6}}}\right)}^{{n}}{}{\mathrm{QPochhammer}}{}\left(\frac{{1}}{{-}{{q}}^{{4}}{+}{{q}}^{{2}}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left(\frac{{1}}{{-}{{q}}^{{5}}{+}{{q}}^{{3}}}{,}{q}{,}{n}\right)}{{\left({2}{}{{q}}^{{2}}{-}{1}\right)}^{{2}}{}{\left({{q}}^{{3}}{+}{1}\right)}^{{2}}{}{\left({{q}}^{{4}}{+}{1}\right)}^{{2}}{}\left({q}{+}{1}\right){}\left({{q}}^{{2}}{+}{1}\right){}\left({{q}}^{{11}}{+}{{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{10}}{+}{{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{9}}{+}{{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{8}}{+}{{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{7}}{+}{{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{6}}{+}{{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{5}}{+}{{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{4}}{+}{{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{3}}{+}{{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{2}}{+}{q}{-}{1}\right){}{\mathrm{QPochhammer}}{}\left({-}{{q}}^{{4}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-}\frac{{1}}{{{q}}^{{5}}}{,}{q}{,}{n}\right)}$ (2)
 > $\mathrm{QEfficientRepresentation}\left[2\right]\left(H,q,n\right)$
 $\frac{{2}{}\left({{q}}^{{5}}{-}{{q}}^{{3}}{+}{1}\right){}\left({{q}}^{{2}}{+}{q}{-}{1}\right){}\left({q}{+}{1}\right){}\left({{q}}^{{2}}{+}{1}\right){}{\left({{q}}^{{4}}{-}{{q}}^{{2}}{+}{1}\right)}^{{2}}{}{\left({{q}}^{{3}}{-}{q}{+}{1}\right)}^{{2}}{}\left({{q}}^{{3}}{+}{{q}}^{{n}}\right){}\left({{q}}^{{4}}{+}{{q}}^{{n}}\right){}{\left(\frac{{\left({{q}}^{{2}}{-}{1}\right)}^{{2}}}{{{q}}^{{6}}}\right)}^{{n}}{}{\mathrm{QPochhammer}}{}\left({-}\frac{{{q}}^{{12}}}{{{q}}^{{2}}{-}{1}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-}\frac{{{q}}^{{3}}}{{{q}}^{{2}}{-}{1}}{,}{q}{,}{n}\right)}{{{q}}^{{5}}{}\left({{q}}^{{4}}{+}{1}\right){}{\left({{q}}^{{3}}{+}{{q}}^{{n}}{-}{q}\right)}^{{2}}{}{\left({{q}}^{{n}}{+}{{q}}^{{4}}{-}{{q}}^{{2}}\right)}^{{2}}{}\left({{q}}^{{n}}{+}\frac{{1}}{{{q}}^{{3}}}\right){}\left({{q}}^{{n}}{+}\frac{{1}}{{{q}}^{{2}}}\right){}\left({{q}}^{{n}}{+}\frac{{1}}{{q}}\right){}\left({{q}}^{{n}}{+}{1}\right){}\left({{q}}^{{n}}{+}\frac{{{q}}^{{2}}{-}{1}}{{q}}\right){}\left({{q}}^{{2}}{+}{{q}}^{{n}}{-}{1}\right){}\left({{q}}^{{n}}{+}{{q}}^{{5}}{-}{{q}}^{{3}}\right){}{\mathrm{QPochhammer}}{}\left({-}\frac{{1}}{{{q}}^{{5}}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-}\frac{{1}}{{{q}}^{{4}}}{,}{q}{,}{n}\right)}$ (3)
 > $\mathrm{QEfficientRepresentation}\left[3\right]\left(H,q,n\right)$
 $\frac{{{q}}^{{2}}{}\left({{q}}^{{3}}{-}{q}{+}{1}\right){}\left({{q}}^{{4}}{-}{{q}}^{{2}}{+}{1}\right){}\left({{q}}^{{n}}{+}\frac{{{q}}^{{2}}{-}{1}}{{{q}}^{{2}}}\right){}{\left({{q}}^{{3}}{+}{{q}}^{{n}}\right)}^{{2}}{}{\left({{q}}^{{4}}{+}{{q}}^{{n}}\right)}^{{2}}{}\left({q}{+}{{q}}^{{n}}\right){}\left({{q}}^{{2}}{+}{{q}}^{{n}}\right){}{\left(\frac{{\left({{q}}^{{2}}{-}{1}\right)}^{{2}}}{{{q}}^{{6}}}\right)}^{{n}}{}{\mathrm{QPochhammer}}{}\left({-}\frac{{{q}}^{{12}}}{{{q}}^{{2}}{-}{1}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left(\frac{{1}}{{-}{{q}}^{{5}}{+}{{q}}^{{3}}}{,}{q}{,}{n}\right)}{\left({2}{}{{q}}^{{2}}{-}{1}\right){}{\left({{q}}^{{3}}{+}{1}\right)}^{{2}}{}{\left({{q}}^{{4}}{+}{1}\right)}^{{2}}{}\left({q}{+}{1}\right){}\left({{q}}^{{2}}{+}{1}\right){}\left({{q}}^{{3}}{+}{{q}}^{{n}}{-}{q}\right){}\left({{q}}^{{n}}{+}{{q}}^{{4}}{-}{{q}}^{{2}}\right){}{\mathrm{QPochhammer}}{}\left({-}{{q}}^{{4}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-}\frac{{1}}{{{q}}^{{5}}}{,}{q}{,}{n}\right)}$ (4)
 > $\mathrm{QEfficientRepresentation}\left[4\right]\left(H,q,n\right)$
 $\frac{{2}{}\left({{q}}^{{4}}{-}{{q}}^{{2}}{+}{1}\right){}\left({{q}}^{{3}}{-}{q}{+}{1}\right){}\left({q}{+}{1}\right){}\left({{q}}^{{2}}{+}{1}\right){}\left({{q}}^{{3}}{+}{{q}}^{{n}}\right){}\left({{q}}^{{4}}{+}{{q}}^{{n}}\right){}\left({{q}}^{{n}}{+}\frac{{{q}}^{{2}}{-}{1}}{{{q}}^{{2}}}\right){}{\left(\frac{{\left({{q}}^{{2}}{-}{1}\right)}^{{2}}}{{{q}}^{{6}}}\right)}^{{n}}{}{\mathrm{QPochhammer}}{}\left(\frac{{1}}{{-}{{q}}^{{5}}{+}{{q}}^{{3}}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-}\frac{{{q}}^{{12}}}{{{q}}^{{2}}{-}{1}}{,}{q}{,}{n}\right)}{{{q}}^{{4}}{}\left({2}{}{{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{4}}{+}{1}\right){}\left({{q}}^{{n}}{+}\frac{{1}}{{{q}}^{{3}}}\right){}\left({{q}}^{{n}}{+}\frac{{1}}{{{q}}^{{2}}}\right){}\left({{q}}^{{n}}{+}\frac{{1}}{{q}}\right){}\left({{q}}^{{n}}{+}{1}\right){}\left({{q}}^{{3}}{+}{{q}}^{{n}}{-}{q}\right){}\left({{q}}^{{n}}{+}{{q}}^{{4}}{-}{{q}}^{{2}}\right){}{\mathrm{QPochhammer}}{}\left({-}\frac{{1}}{{{q}}^{{4}}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-}\frac{{1}}{{{q}}^{{5}}}{,}{q}{,}{n}\right)}$ (5)
 > $\mathrm{RegularQPochhammerForm}\left(H,q,n\right)$
 $\frac{{\left(\frac{{\left({{q}}^{{2}}{-}{1}\right)}^{{2}}}{{{q}}^{{6}}}\right)}^{{n}}{}{\mathrm{QPochhammer}}{}\left({-}\frac{{{q}}^{{3}}}{{{q}}^{{2}}{-}{1}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left(\frac{{1}}{{-}{{q}}^{{4}}{+}{{q}}^{{2}}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-}\frac{{{q}}^{{12}}}{{{q}}^{{2}}{-}{1}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-}\frac{{1}}{{{q}}^{{2}}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left(\frac{{1}}{{-}{{q}}^{{5}}{+}{{q}}^{{3}}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-1}{,}{q}{,}{n}\right)}{{\mathrm{QPochhammer}}{}\left({-}\frac{{1}}{{{q}}^{{5}}}{,}{q}{,}{n}\right){}{{\mathrm{QPochhammer}}{}\left({-}\frac{{1}}{{{q}}^{{4}}}{,}{q}{,}{n}\right)}^{{2}}{}{\mathrm{QPochhammer}}{}\left(\frac{{1}}{{-}{{q}}^{{2}}{+}{1}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-}{{q}}^{{4}}{,}{q}{,}{n}\right){}{\mathrm{QPochhammer}}{}\left({-}\frac{{{q}}^{{2}}}{{{q}}^{{2}}{-}{1}}{,}{q}{,}{n}\right)}$ (6)

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.