  Given arbitrary homogeneous ideals $I$ and $J$ in polynomial rings $A$ and
$B$ over a field $k$, we investigate the depth and the Castelnuovo-Mumford
regularity of powers of the sum $I+J$ in $A \otimes_k B$ in terms of those of
$I$ and $J$. Our results can be used to study the behavior of the depth and
regularity functions of powers of an ideal. For instance, we show that such a
depth function can take as its values any infinite non-increasing sequence of
non-negative integers.
