Rational digit systems over finite fields and Christol's Theorem

Let P,Q∈Fq[X]∖{0} be two coprime polynomials over the finite field Fq with deg⁡P>deg⁡Q. We represent each polynomial w over Fq byw=∑i=0ksiQ(PQ)i using a rational base P/Q and digits si∈Fq[X] satisfying deg⁡si