Hyperquadratic continued fractions in odd characteristic with partial quotients of degree one

In 1986, some examples of algebraic, and nonquadratic, power series over a finite prime field, having a continued fraction expansion with partial quotients all of degree 1 were discovered by W. Mills and D. Robbins. In this note we show that these few examples belong to a very large family of continued fractions for certain algebraic power series over an arbitrary finite field of odd characteristic.