Q:

Explicit Fractional Basis Expansion

We can write any $x\in\mathbb R$ as its $b>1\in \mathbb N$ basis expansion:
$$x=sgn(x)\sum_{d=-\infty}^\infty b^d \lfloor |x|b^{-d} - b\lfloor |x|b^{-d-1} \rfloor \rfloor$$
Can one come with the same kind of explicit expansion that would be valid for any rational $b=p/q$? 

A:

Yes; this is known as Non-integer representation or $\beta$-expansion. 