  We present a new quantum field-theoretic definition of fully unintegrated
dihadron fragmentation functions (DiFFs) as well as a generalized version for
$n$-hadron fragmentation functions. We demonstrate that this definition allows
certain sum rules to be satisfied, making it consistent with a number density
interpretation. Moreover, we show how our corresponding so-called extended
DiFFs that enter existing phenomenological studies are number densities and
also derive their evolution equations. Within this new framework, DiFFs
extracted from experimental measurements will have a clear physical meaning.
