On the Order of Polynilpotent Multipliers of Some Nilpotent Products of Cyclic $p$-Groups

In this article we show that if $\mathcal{V}$ is the variety of polynilpotent groups of class row $(c_1,c_2,\mathrm{.}\mathrm{.}\mathrm{.},c_s),{\mathcal{N}}_{c_1,c_2,\mathrm{.}\mathrm{.}\mathrm{.},c_s}$, and $G\cong{\bf {Z}}{p^{\alpha_1}}\stackrel{n}{*}{\bf {Z}}{p^{\alpha_2}}\stackrel{n}{}…\stackrel{n}{}{\bf{Z}}{p^{\alpha_t} }$isthe$n$thnilpotentproductofsomecyclic$p$-groups,where$c_1\geq n$,$\alpha_1 \geq \alpha_2 \geq…\geq \alpha_t $and$ (q,p)=1$forallprimes$q$lessthanorequalto$n$,then$|{\mathcal N}{c_1,c_2,…,c_s}M(G)|=p^{d_m}$ifandonlyif$G\cong{\bf {Z}}{p}\stackrel{n}{*}{\bf {Z}}{p}\stackrel{n}{}…\stackrel{n}{}{\bf{Z}}{p }$($m$-copies),where$m=\sum {i=1}^t \alpha_i$and$d_m=\chi{c_s+1}(…(\chi{c_2+1}(\sum_{j=1}^n \chi_{c_1+j}(m)))…)$. Also, we extend the result to the multiple nilpotent product $G\cong{\bf {Z}}{p^{\alpha_1}}\stackrel{n_1}{*}{\bf {Z}}{p^{\alpha_2}}\stackrel{n_2}{}…\stackrel{n_{t-1}}{}{\bf{Z}}{p^{\alpha_t} }$,where$c_1\geq n_1\geq…\geq n{t-1}$. Finally a similar result is given for the $c$-nilpotent multiplier of $G\cong{\bf {Z}}{p^{\alpha_1}}\stackrel{n}{*}{\bf {Z}}{p^{\alpha_2}}\stackrel{n}{}…\stackrel{n}{}{\bf{Z}}_{p^{\alpha_t}}$withthedifferentconditions$n \geq c$and$ (q,p)=1$forallprimes$q$ less than or equal to $n+c\mathrm{.}$