Q-shifted factorial

The $q$-shifted factorial $(a;q)_n$ is defined for $a,q \in \mathbb{C}$ by $(a;q)_0=1$ and for $n=1,2,3,\ldots$ or $n=\infty$, by $$(a;q)_n=\left\{ \begin{array}{ll} 1, & \quad n=0 \\ \displaystyle\prod_{k=0}^{n-1} 1-aq^{k}=(1-a)(1-aq)(1-aq^2)\ldots(1-aq^{n-1}), & \quad n=1,2,3,\ldots \\ \end{array} \right.$$

References

• 1953: Harry Bateman: Higher Transcendental Functions Volume I ... (previous) ... (next): $4.8 (1) (2)$ (assumes $|q|<1$)
• Daniel S. Moak: The q-gamma function for q greater than 1 (1980)... (previous)... (next)
• Tom H. Koornwinder: q-Special functions, a tutorial (1994)... (previous)... (next)
• 1999: George E. Andrews, Richard Askey and Ranjan Roy: Special Functions ... (previous) ... (next) $(10.2.1)$