On nilpotent multipliers of some verbal products of groups

The paper is devoted to finding a homomorphic image for the c-nilpotent multiplier of the verbal product of a family of groups with respect to a variety V when V⊆Nc or Nc⊆V. Also a structure of the c-nilpotent multiplier of a special case of the verbal product, the nilpotent product, of cyclic groups is given. In fact, we present an explicit formula for the c-nilpotent multiplier of the nth nilpotent product of the group , where ri+1 divides ri for all i, 1⩽i⩽t−1, and (p,r1)=1 for any prime p less than or equal to n+c, for all positive integers n, c.