I just came to know that it is possible to define a generalized base as an infinite sequence of natural numbers $\mathbf b=(b_1,b_2,\dots)$ where $b_i\ge 2$ for all $i$. With this definition, any $m\in \mathbb N$ can be uniquely expressed as $$m=\overline{x_n\dots x_1}^\mathbf b := x_1 + x_2\cdot b_1 + x_3\cdot b_1b_2 + \dots + x_n\cdot b_1b_2\dots b_{n-1}\tag{*}$$ for some $0\le x_i<b_i$ with $x_n\neq 0$.
First of all, I want to know if there are any proper references where I can read about this generalized base a little bit more.
Secondly, are there more generalized bases known? In other words, given an infinite sequence of natural numbers $\mathbf B=(B_1,B_2,\dots)$, do we know of any conditions on $\mathbf B$ such that any $m\in \mathbb N$ admits a unique representation as $$m=\overline{x_n\dots x_1}^\mathbf B := x_1 + x_2\cdot B_1 + x_3\cdot B_2 + \dots + x_n\cdot B_{n-1}$$ for the $x_i$'s satisfying some relation with respect to the $B_i$'s?
In particular, if we change the condition of $b_i\ge 2$ in $(*)$ to $b_i\ge 1$, then how much of it is still true?
Please also try to provide proper references.
