Integral closures of powers of sums of ideals

Let $k$ be a field, let $A$ and $B$ be polynomial rings over $k$, and let $S=A{\otimes }_{k}B$. Let $I\subseteq A$ and $J\subseteq B$ be monomial ideals. We establish a binomial expansion for rational powers of $I+J\subseteq S$ in terms of those of $I$ and $J$. Particularly, for a positive rational number $u$, we prove that $(I+J{)}_u={\sum }_{0\le \omega \le u,\omega \in \mathbb{Q}}{I}_{\omega }{J}_{u-\omega },$ and that the sum on the right hand side is a finite sum. This finite sum can be made more precise using jumping numbers of rational powers of $I$ and $J$. We further give sufficient conditions for this formula to hold for the integral closures of powers of $I+J$ in terms of those of $I$ and $J$. Under these conditions, we provide explicit formulas for the depth and regularity of $\overline{(I+J{)}^k}$ in terms of those of powers of $I$ and $J$.