Expansions in Cantor real bases

We introduce and study series expansions of real numbers with an arbitrary Cantor base $$\varvec{\beta }=(\beta _n)_{n\in {\mathbb {N}}}$$ , which we call }$$ -representations. In doing so, generalize both representations in bases through series. show fundamental properties -representations, each extends existing results on a base. particular, prove generalization Parry’s theorem characterizing sequences nonnegative integers that are the greedy -representations some number interval [0, 1). pay special attention to periodic bases, alternate bases. this case, -shift is sofic if only all quasi-greedy }^{(i)}$$ -expansions 1 ultimately periodic, where i-th shift .