q-Pochhammer symbol

Remarks

For $\abs{q} < 1$ the product defining $(a; q)_\infty$ converges, since $\sum_k \abs{a q^k} = \abs{a} / (1 - \abs{q}) < \infty$; it is zero exactly when some factor is zero. Example: $(q; q)_2 = (1 - q)(1 - q^2) = 1 - q - q^2 + q^3$, by the definition with $a = q$ and $n = 2$. Non-example: for $\abs{q} > 1$ and $a \ne 0$, the factors $1 - a q^k$ do not tend to $1$, so the infinite product does not converge and $(a; q)_\infty$ is not defined. The name comes from the limit $\frac{(q^x; q)_n}{(1 - q)^n} \to x(x+1)\cdots(x+n-1)$ as $q \to 1$, the rising factorial or Pochhammer symbol. The generating function for partitions reads $\sum_{n \ge 0} p(n) q^n = 1/(q; q)_\infty$ for $\abs{q} < 1$.