Binomial expansion and the v-number

Let I⊂A and J⊂B be two monomial ideals, where A and B are two polynomial rings with disjoint variables. Considering a general set-up of monomial filtrations, we study the behavior of the v-number under binomial expansion. As an application, we get an explicit formula of v((I+J)(k)) in terms of v(I(i)) and v(J(j)), where L(k) denotes the k-th symbolic power of an ideal L. Furthermore, an analogous formula is extended for the v-number of integral closure of (I+J)k.