On the convergence of periodic continued fractions via linearly recurring sequences

Abstract

In this paper, we give a new process to study the convergence of periodic continued fractions, making use of the approach on the linear difference equations with periodic coefficients, based on the analytic formula of the linearly recurring sequences. Some new explicit expressions of nth numerators and nth denominators of the convergents of periodic continued fractions lead to discuss their convergence. Moreover, we provide a new discussion on the closed link between quadratic irrational numbers and simple periodic continued fractions. Some applications of the Pell’s equations are given and some numerical examples which illustrate the theory are supplied.