Periodic representations and rational approximations for quadratic irrationalities by means of Rédei rational functions

Thepaperis devoted to the problem of representing and approximating quadratic irrationalities. In particular, a new manageable periodic representation of period 2 and pre-period 1 is provided, for any quadratic irrationality, by means of continued fractions with rational partial quotients, considering appropriate Rédei rational functions. These representations lead to infinitely many different kinds of rational approximations for the represented irrational. We prove that among these approximations, we can find the Newton approximations, providing a fast way to evaluate them using powers of matrices. Finally, we see that our periodic continued fractions with rational partial quotients easily transform under linear fractional transformations.