GENERALIZED CONTINUED FRACTIONS IN REAL QUADRATIC FIELDS AND PELL'S EQUATIONS

The central objects of study of this paper are the lambda-fractions (or Rosen fractions) introduced by Rosen in [5] and studied further in [6] and [7]. These expressions relax the restriction on classical continued fractions that all partial quotients be positive integers to allow for partial quotients of the form r lambda, where r is a positive integer and. is a fixed algebraic integer. The original context in which these expressions arose was the consideration of cusps in certain Fuchsian groups where the considered values of lambda considered were determined by the relevant geometry. Our study restricts to the continued fraction expressions of real quadratic surds by taking lambda = root d for positive square- free integer d. The main result of this paper is a complete classification of the generalized continued fraction expression of units in a real quadratic field via the recursion relation arising in solving a generalized Pell equation.