\[(1-x)\sum_{m,n=0}^{\infty}p_{m}\left(\leq k,\mathcal{D},n\right)x^{m}q^{n}=%
\sum_{m=0}^{k}\genfrac{[}{]}{0.0pt}{}{k}{m}_{q}q^{m(m+1)/2}x^{m}=\prod_{j=1}^{%
k}(1+x\,q^{j}),\]