  In this paper, we further investigate properties of generalized bent Boolean
functions from $\Z_{p}^n$ to $\Z_{p^k}$, where $p$ is an odd prime and $k$ is a
positive integer. For various kinds of representations, sufficient and
necessary conditions for bent-ness of such functions are given in terms of
their various kinds of component functions. Furthermore, a subclass of gbent
functions corresponding to relative difference sets, which we call
$\Z_{p^k}$-bent functions, are studied. It turns out that $\Z_{p^k}$-bent
functions correspond to a class of vectorial bent functions, and the property
of being $\Z_{p^k}$-bent is much stronger then the standard bent-ness. The dual
and the generalized Gray image of gbent function are also discussed. In
addition, as a further generalization, we also define and give
characterizations of gbent functions from $\Z_{p^l}^n$ to $\Z_{p^k}$ for a
positive integer $l$ with $l<k$.
