The commands QSimpComb and QSimplify are for simplification of expressions involving q-hypergeometric terms. For a function $f\left(q^k\right)$, the main use of QSimpComb is for detecting if $f\left(q^k\right)$ is a q-hypergeometric term in $q^k$. That is, if $\frac{f\left(q^{k+1}\right)}{f\left(q^k\right)}$ is a rational function in $q^k$ (see IsQHypergeometricTerm). If the result is not a rational function, QSimplify returns in general a more compact answer.

This implementation is mainly based on the implementation by H. Boeing, W. Koepf. See the Reference Section.

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

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

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

$\mathrm{QSimpComb}\left(\frac{\mathrm{subs}\left(n=n+1\,H\right)}{H}\right)$

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

$\mathrm{IsQHypergeometricTerm}\left(H\,n\,q^n=N\right)$

$\mathrm{true}$

$f≔\mathrm{QPochhammer}\left(a{q}^{-kn}\,q\,n\right)-\frac{\mathrm{QPochhammer}\left(\frac{q}{a}\,q\,kn\right)}{\mathrm{QPochhammer}\left(\frac{q}{a}\,q\,kn-n\right)}{\left(-a\right)}^n{q}^{\mathrm{binomial}\left(n\,2\right)-k{n}^2}$

$f≔\mathrm{QPochhammer}\left(a{q}^{-kn}\,q\,n\right)-\frac{\mathrm{QPochhammer}\left(\frac{q}{a}\,q\,kn\right){\left(-a\right)}^n{q}^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)-k{n}^2}}{\mathrm{QPochhammer}\left(\frac{q}{a}\,q\,kn-n\right)}$

$\mathrm{QSimplify}\left(f\right)$

$0$

$f≔\frac{1}{\mathrm{QPochhammer}\left(a\,q\,2n\right)}\mathrm{QPochhammer}\left(a\,q^2\,n\right)\mathrm{QPochhammer}\left(aq\,q^2\,n\right)$

$f≔\frac{\mathrm{QPochhammer}\left(a\,q^2\,n\right)\mathrm{QPochhammer}\left(qa\,q^2\,n\right)}{\mathrm{QPochhammer}\left(a\,q\,2n\right)}$

$\mathrm{QSimpComb}\left(f\right)$

$\frac{\mathrm{QPochhammer}\left(a\,q^2\,n\right)\mathrm{QPochhammer}\left(qa\,q^2\,n\right)}{\mathrm{QPochhammer}\left(a\,q\,2n\right)}$

$\mathrm{QSimplify}\left(f\right)$

$1$

Boeing, H., and Koepf, W. "Algorithms for q-hypergeometric summation in computer algebra." Journal of Symbolic Computation. Vol. 11. (1999): 1-23.