Second q-shifted factorial

The $q$-shifted factorial $\langle a;q \rangle_n$ (not to be confused with the $q$-Pochhammer $(a;q)_n$) is given by $$\langle a;q \rangle_n = \left\{ \begin{array}{ll} 1, & n=0; \\ \displaystyle\prod_{k=0}^{n-1} (1-q^{a+m}), & n=1,2,\ldots \end{array} \right.$$ If $(a)=(a_1,a_2,\ldots,a_m)$ is a vector then we define the notation $$\langle (a);q \rangle_n = \langle a_1,a_2,\ldots,a_m; q \rangle_n = \displaystyle\prod_{j=1}^m \langle a_j;q \rangle_n.$$