$$1+z+...+z^{k-1}=\prod_{l=1}^{i}{\prod_{r=1}^{q_l}{(1+z^{p_l^{r-1}\prod\limits_{m=1}^{l-1}{p_m^{q_m}}}+...+z^{p_l^{r-1}(p_l-1)\prod\limits_{m=1}^{l-1}{p_m^{q_m}}})}} (3.1)$$