The QDifferenceEquations package supports five q-hypergeometric terms. They are q-Pochhammer symbol, q-binomial coefficient, q-brackets, q-factorial, and q-Gamma, which correspond to the five functions QPochhammer, QBinomial, QBrackets, QFactorial, and QGAMMA.

These functions are placeholders for the q-objects. The command expand allows expansion of these objects. The command $\mathrm{convert}\left(...,\mathrm{QPochhammer}\right)$ allows the re-write of QBinomial, QBrackets, QFactorial, and QGAMMA in terms of QPochhammer symbols.

The five q-hypergeometric objects are defined as follows.

Note that $\mathrm{QPochhammer}\left(\mathrm{seq}\left(a_i\,i=1..n\right)\,q\,k\right)$ (the compact Gasper and Rahman notation) means $\prod _{i=1}^n\mathrm{QPochhammer}\left(a_i\,q\,k\right)$.

The commands QSimpComb and QSimplify are for simplification of expressions involving these q-objects.

This implementation is mainly based on the implementation by H. Boeing, W. Koepf. See the References section.

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

$\mathrm{expand}\left(\mathrm{QPochhammer}\left(a\,q\,4\right)\right)$

$\left(1-a\right)\left(-aq+1\right)\left(-a{q}^2+1\right)\left(-a{q}^3+1\right)$

$\mathrm{expand}\left(\mathrm{QPochhammer}\left(a\,q\,-4\right)\right)$

$\frac{1}{\left(1-\frac{a}{q^4}\right)\left(1-\frac{a}{q^3}\right)\left(1-\frac{a}{q^2}\right)\left(1-\frac{a}{q}\right)}$

$\mathrm{expand}\left(\mathrm{QBrackets}\left(k\,q\right)\right)$

$\frac{q^k-1}{q-1}$

$\mathrm{convert}\left(\mathrm{QBinomial}\left(n\,k\,q\right)\,'\mathrm{QPochhammer}'\right)$

$\frac{\mathrm{QPochhammer}\left(q\,q\,n\right)}{\mathrm{QPochhammer}\left(q\,q\,k\right)\mathrm{QPochhammer}\left(q\,q\,n-k\right)}$

$\mathrm{convert}\left(\mathrm{QGAMMA}\left(z\,q\right)\,'\mathrm{QPochhammer}'\right)$

$\frac{\mathrm{QPochhammer}\left(q\,q\,\mathrm{\infty }\right){\left(1-q\right)}^{1-z}}{\mathrm{QPochhammer}\left(q^z\,q\,\mathrm{\infty }\right)}$

$\mathrm{convert}\left(\mathrm{QFactorial}\left(k\,q\right)\,'\mathrm{QPochhammer}'\right)$

$\frac{\mathrm{QPochhammer}\left(q\,q\,k\right)}{{\left(1-q\right)}^k}$

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

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

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

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

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