$$\frac{\left(q;q\right)_{a_{1}+a_{2}+\cdots+a_{n}}}{\left(q;q\right)_{a_{1}}\left(q;q\right)_{a_{2}}\cdots\left(q;q\right)_{a_{n}}}=\mbox{ coeff. of }x_{1}^{0}x_{2}^{0}\cdots x_{n}^{0}\mbox{ in }\prod_{1\leq j<k\leq n}\left(\frac{x_{j}}{x_{k}};q\right)_{a_{j}}\left(\frac{qx_{k}}{x_{j}};q\right)_{a_{k}}$$