Characterization of generalized bent functions of algebraic degree $1$

Bent functions of the form $\mathbb{F}_2^n\rightarrow\mathbb{Z}_q$, where $q\geqslant2$ is a positive integer, are known as generalized bent (gbent) functions. A gbent function for which it is possible to define a dual gbent function is called regular. We study gbent functions of degree $1$. Criterion of the generalized Boolean function of degree $1$ to be gbent is obtained. The conditions under which the function is regular or weakly regular are described. Component Boolean functions are investigated, it follows that for the case $q=2^k$ two of them, having maximal indices, are quadratic, while the rest are constant.