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.