Q-Pochhammer symbol

In the mathematical field of combinatorics, the q-Pochhammer symbol, also called the q-shifted factorial, is the product $${\displaystyle (a;q)_{n}=\prod _{k=0}^{n-1}(1-aq^{k})=(1-a)(1-aq)(1-aq^{2})\cdots (1-aq^{n-1}),}$$ with ${\displaystyle (a;q)_{0}=1.}$ It is a q-analog of the Pochhammer symbol ${\displaystyle (x)_{n}=x(x+1)\dots (x+n-1)}$, in the sense that $${\displaystyle \lim _{q\to 1}{\frac {(q^{x};q)_{n}}{(1-q)^{n}}}=(x)_{n}.}$$ The q-Pochhammer symbol is a major building block in the construction of q-analogs; for instance, in the theory of basic hypergeometric series, it plays the role that the ordinary Pochhammer symbol plays in the theory of generalized hypergeometric series.

Unlike the ordinary Pochhammer symbol, the q-Pochhammer symbol can be extended to an infinite product: $${\displaystyle (a;q)_{\infty }=\prod _{k=0}^{\infty }(1-aq^{k}).}$$ This is an analytic function of q in the interior of the unit disk, and can also be considered as a formal power series in q. The special case $${\displaystyle \phi (q)=(q;q)_{\infty }=\prod _{k=1}^{\infty }(1-q^{k})}$$ is known as Euler's function, and is important in combinatorics, number theory, and the theory of modular forms.