Let be a finite field and

with
.
We define a new kind of digit system in the quotient ring
.
There are striking analogies of these digit systems to the well
known canonical number systems (CNS) defined in
for a polynomial
.
Especially, the problem of characterizing the possible bases is solved.
In contrast, such a characterization for CNS is still outstanding.
Furthermore, we will study -expansions in
,
the field of formal Laurent series over in the variable . Once again,
we can solve problems corresponding to open problems on classical
-expansions.
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Please send comments and corrections to Thomas Klausner.