This article is devoted to present an explicit formula for the $c$th nilpotent multiplier of nilpotent products of some cyclic groups $G={ \bf { Z}}\stackrel{ n_1}{ *}{ \bf { Z}}\stackrel{ n_2}{ *}...\stackrel{ n_{ t-1}}{ *}{ \bf { Z}}\stackrel{ n_{ t}}{ *} { \bf { Z}}_{ m_{ t+1}}\stackrel{ n_{ t+1}}{ *}{ \bf { Z}}_{ m_{ t+2}}\stackrel{ n_{ t+2}}{ *}...\stackrel{ n_{ k}}{ *}{ \bf{ Z}}_{ m_{ k+1}}$, where $m_{ i+1} | m_i$ for all $t+1 \leq i \leq k$ and $c \geq n_1\geq n_2\geq ...\geq n_t\geq ...\geq n_{ k}$ such that $ (p,m_{ t+1})=1$ for all prime $p \leq n_1$. Moreover, we compute the polynilpotent multiplier of the group $G$ with respect to the polynilpotent variety ${ \mathcal N}_{ c_1,c_2,...,c_s}$, where $c_1 \geq n_1.$
Categories: math.GR