Continued fractions for hyperquadratic power series over a finite field

An irrational power series over a finite field Fq of characteristic p is called hyperquadratic if it satisfies an algebraic equation of the form x=(Axr+B)/(Cxr+D), where r is a power of p and the coefficients belong to Fq[T]. These algebraic power series are analogues of quadratic real numbers. This analogy makes their continued fraction expansions specific as in the classical case, but more sophisticated. Here we present a general result on the way some of these expansions are generated. We apply it to describe several families of expansions having a regular pattern.