Polynilpotent Multipliers of Some Nilpotent Products of Cyclic Groups

In this article, we present an explicit formula for the $c$th nilpotent multiplier (the Baer invariant with respect to the variety of nilpotent groups of class at most $c\ge 1$) of the $n$th nilpotent product of some cyclic groups $G={\mathbb {Z}}\stackrel{n}{} … \stackrel{n}{}{\mathbb {Z}}\stackrel{n}{} {\mathbb {Z}}_{r_1}\stackrel{n}{} … \stackrel{n}{*}{$\mathbb{Z}$}{r_t}$,(m-copiesof$\mathbb {Z}$),where$r{i+1} | r_i$for$1 \leq i \leq t-1$and$c \geq n$suchthat$ (p,r_1)=1$ for all primes $p$ less than or equal to $n$. Also, we compute the polynilpotent multiplier of the group $G$ with respect to the polynilpotent variety ${\mathcal{N}}_{c_1,c_2,\mathrm{.}\mathrm{.}\mathrm{.},c_t}$, where $c_1\ge n\mathrm{.}$