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 behaviour of the v-function 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) denote the symbolic power of an ideal L. Furthermore, an analogous formula is extended for the v-function of integral closure of (I+J) ᵏ.