Let
P,Q∈Fq[X]∖{0} be two coprime polynomials over the finite field
Fq with
ac6b75697f2" title="Click to view the MathML source">degP>degQ. We represent e
ach polynomial
w over
Fq by
using a rational
base P/Q and
digits ac2d9c567c3d57a4aed0d93ac" title="Click to view the MathML source">si∈Fq[X] satisfying
degsi<degP.
Digit expansions of this type are also defined 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 char
acterize 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.