Expansions in Cantor real bases

We introduce and study series expansions of real numbers with an arbitrary Cantor real base. In doing so, we generalize both representations of real numbers in real bases and through Cantor series. We show fundamental properties of these new representations, each of which extends existing results on representations in a real base. In particular, we prove a generalization of Parry's theorem characterizing sequences of nonnegative integers that are the greedy representations of some real number in the interval [0,1). We pay special attention to periodic Cantor real bases, which we call alternate bases. In this case, we show that the associated shift is sofic if and only if all quasi-greedy shifted expansions of 1 are ultimately periodic.