${\mathbb{Z}}_q$-valued generalized bent functions in odd characteristics

In this paper, we investigate properties of functions from ${\mathbb{Z}}_p^n$ to ${\mathbb{Z}}_q$, where $p$ is an odd prime and $q$ is a positive integer divided by $p$. we present the sufficient and necessary conditions for bent-ness of such generalized Boolean functions in terms of classical $p$-ary bent functions, when $q=p^k$. When $q$ is divided by $p$ but not a power of it, we give an sufficient condition for weakly regular gbent functions. Some related constructions are also obtained.