In 2017, Tang et al. provided a complete characterization of generalized bent functions from ℤ2n to ℤq(q = 2m) in terms of their component functions (IEEE Trans. Inf. Theory. vol.63, no.7, pp.4668-4674). In this letter, for a general even q, we aim to provide some characterizations and more constructions of generalized bent functions with flexible coefficients. Firstly, we present some sufficient conditions for a generalized Boolean function with at most three terms to be gbent. Based on these results, we give a positive answer to a remaining question proposed by Hodžić in 2015. We also prove that the sufficient conditions are also necessary in some special cases. However, these sufficient conditions whether they are also necessary, in general, is left as an open problem. Secondly, from a uniform point of view, we provide a secondary construction of gbent function, which includes several known constructions as special cases.

In this paper, a new class of bent functions is constructed by combining class M and class C bent functions. Using the construction method of the class D bent functions defined on the binary vector space, new p-ary generalized bent functions are also introduced for odd prime p.

In this paper, using p-ary bent functions defined on vector space over the finite field Fpk, we generalized the construction method of the families of p-ary bent sequences with balanced and optimal correlation properties introduced by Kumar and Moreno for an odd prime p, called generalized p-ary bent sequences. It turns out that the family of balanced p-ary sequences with optimal correlation property introduced by Moriuchi and Imamura is a special case of the newly constructed generalized p-ary bent sequences.