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,

$Q\left(n\right)$ has the minimal number of factors,

$V\left(q^n\right)$ 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.

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

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

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

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

$\frac{q^{66}{\left(q^n+\frac{q^2-1}{q^2}\right)}^2{\left(q^3+q^n\right)}^2{\left(q^4+q^n\right)}^2\left(q+q^n\right)\left(q^2+q^n\right)\left(q^n+\frac{q^2-1}{q^{11}}\right)\left(q^n+\frac{q^2-1}{q^{10}}\right)\left(q^n+\frac{q^2-1}{q^9}\right)\left(q^n+\frac{q^2-1}{q^8}\right)\left(q^n+\frac{q^2-1}{q^7}\right)\left(q^n+\frac{q^2-1}{q^6}\right)\left(q^n+\frac{q^2-1}{q^5}\right)\left(q^n+\frac{q^2-1}{q^4}\right)\left(q^n+\frac{q^2-1}{q^3}\right)\left(q^n+\frac{q^2-1}{q}\right)\left(q^2+q^n-1\right){\left(\frac{{\left(q^2-1\right)}^2}{q^6}\right)}^n\mathrm{QPochhammer}\left(\frac{1}{-q^4+q^2}\,q\,n\right)\mathrm{QPochhammer}\left(\frac{1}{-q^5+q^3}\,q\,n\right)}{{\left(2q^2-1\right)}^2{\left(q^3+1\right)}^2{\left(q^4+1\right)}^2\left(q+1\right)\left(q^2+1\right)\left(q^{11}+q^2-1\right)\left(q^{10}+q^2-1\right)\left(q^9+q^2-1\right)\left(q^8+q^2-1\right)\left(q^7+q^2-1\right)\left(q^6+q^2-1\right)\left(q^5+q^2-1\right)\left(q^4+q^2-1\right)\left(q^3+q^2-1\right)\left(q^2+q-1\right)\mathrm{QPochhammer}\left(-q^4\,q\,n\right)\mathrm{QPochhammer}\left(-\frac{1}{q^5}\,q\,n\right)}$

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

$\frac{2\left(q^5-q^3+1\right)\left(q^2+q-1\right)\left(q+1\right)\left(q^2+1\right){\left(q^4-q^2+1\right)}^2{\left(q^3-q+1\right)}^2\left(q^3+q^n\right)\left(q^4+q^n\right){\left(\frac{{\left(q^2-1\right)}^2}{q^6}\right)}^n\mathrm{QPochhammer}\left(-\frac{q^{12}}{q^2-1}\,q\,n\right)\mathrm{QPochhammer}\left(-\frac{q^3}{q^2-1}\,q\,n\right)}{q^5\left(q^4+1\right){\left(q^3+q^n-q\right)}^2{\left(q^n+q^4-q^2\right)}^2\left(q^n+\frac{1}{q^3}\right)\left(q^n+\frac{1}{q^2}\right)\left(q^n+\frac{1}{q}\right)\left(q^n+1\right)\left(q^n+\frac{q^2-1}{q}\right)\left(q^2+q^n-1\right)\left(q^n+q^5-q^3\right)\mathrm{QPochhammer}\left(-\frac{1}{q^4}\,q\,n\right)\mathrm{QPochhammer}\left(-\frac{1}{q^5}\,q\,n\right)}$

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

$\frac{q^2\left(q^3-q+1\right)\left(q^4-q^2+1\right){\left(q^3+q^n\right)}^2{\left(q^4+q^n\right)}^2\left(q+q^n\right)\left(q^2+q^n\right)\left(q^n+\frac{q^2-1}{q^2}\right){\left(\frac{{\left(q^2-1\right)}^2}{q^6}\right)}^n\mathrm{QPochhammer}\left(\frac{1}{-q^5+q^3}\,q\,n\right)\mathrm{QPochhammer}\left(-\frac{q^{12}}{q^2-1}\,q\,n\right)}{{\left(q^3+1\right)}^2{\left(q^4+1\right)}^2\left(q+1\right)\left(q^2+1\right)\left(2q^2-1\right)\left(q^3+q^n-q\right)\left(q^n+q^4-q^2\right)\mathrm{QPochhammer}\left(-q^4\,q\,n\right)\mathrm{QPochhammer}\left(-\frac{1}{q^5}\,q\,n\right)}$

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

$\frac{2\left(q^4-q^2+1\right)\left(q^3-q+1\right)\left(q+1\right)\left(q^2+1\right)\left(q^3+q^n\right)\left(q^4+q^n\right)\left(q^n+\frac{q^2-1}{q^2}\right){\left(\frac{{\left(q^2-1\right)}^2}{q^6}\right)}^n\mathrm{QPochhammer}\left(-\frac{q^{12}}{q^2-1}\,q\,n\right)\mathrm{QPochhammer}\left(\frac{1}{-q^5+q^3}\,q\,n\right)}{q^4\left(2q^2-1\right)\left(q^4+1\right)\left(q^n+\frac{1}{q^3}\right)\left(q^n+\frac{1}{q^2}\right)\left(q^n+\frac{1}{q}\right)\left(q^n+1\right)\left(q^3+q^n-q\right)\left(q^n+q^4-q^2\right)\mathrm{QPochhammer}\left(-\frac{1}{q^5}\,q\,n\right)\mathrm{QPochhammer}\left(-\frac{1}{q^4}\,q\,n\right)}$

$\mathrm{RegularQPochhammerForm}\left(H\,q\,n\right)$

$\frac{{\left(\frac{{\left(q^2-1\right)}^2}{q^6}\right)}^n\mathrm{QPochhammer}\left(-\frac{q^{12}}{q^2-1}\,q\,n\right)\mathrm{QPochhammer}\left(\frac{1}{-q^5+q^3}\,q\,n\right)\mathrm{QPochhammer}\left(\mathrm{-1}\,q\,n\right)\mathrm{QPochhammer}\left(-\frac{1}{q^2}\,q\,n\right)\mathrm{QPochhammer}\left(\frac{1}{-q^4+q^2}\,q\,n\right)\mathrm{QPochhammer}\left(-\frac{q^3}{q^2-1}\,q\,n\right)}{\mathrm{QPochhammer}\left(-q^4\,q\,n\right)\mathrm{QPochhammer}\left(-\frac{q^2}{q^2-1}\,q\,n\right){\mathrm{QPochhammer}\left(-\frac{1}{q^4}\,q\,n\right)}^2\mathrm{QPochhammer}\left(\frac{1}{-q^2+1}\,q\,n\right)\mathrm{QPochhammer}\left(-\frac{1}{q^5}\,q\,n\right)}$

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.