On continued fraction expansions of quadratic irrationals in positive characteristic

Let $P$ be a prime polynomial in the variable $Y$ over a finite field and let $f$ be a quadratic irrational in the field of formal Laurant series in the variable $Y^{-1}$. We study the asymptotic properties of the degrees of the coefficients of the continued fraction expansion of quadratic irrationals such as $P^{n}f$ and prove results that are in sharp contrast to the analogue situation in zero characteristic.