On a quartic equation and two families of hyperquadratic continued fractions in pwer series fields

Casually introduced thirty years ago, a simple algebraic equation of degree 4 with coefficients in F_p[T] has a solution in the field of power series over the finite field F_p. For each p>3, the continued fraction expansion of this solution is remarkable and it has a different general pattern according to the remainder, 1 or 2, in the division of p by 3. We describe two very large families of algebraic continued fractions, each containing these solutions, according to the class of p modulo 3. We compute the irrationality measure for these algebraic continued fractions, and as a consequence, we obtain two different values for the solution of the quartic equation, only depending on the class of p modulo 3