Let
P,Q∈Fq[X]∖{0} be two coprime polynomials over the
finite field
Fq with
degP>degQ. We represent each polynomial
w over
Fq by
using a rational
base P/Q and
digits si∈Fq[X] satisfying
degsi<degP.
Digit expansions of this type are also de
fined for formal Laurent series over
Fq. We prove uniqueness and automatic properties of these expansions. Although the
ω -language of the possible digit strings is not regular, we are able to characterize the digit expansions of algebraic elements. In particular, we give a version of Christol's Theorem by showing that the digit string of the digit expansion of a formal Laurent series is automatic if and only if the series is algebraic over
Fq[X]. Finally, we study relations between digit expansions of formal Laurent series and a
finite fields version of Mahler's 3/2-problem.