The central objects of study of this paper are the l-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 rl, where r is a positive integer and l 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 l considered were determined by the relevant geometry. Our study restricts to the continued fraction expressions of real quadratic surds by taking ?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.

continued fractions, Rosen fractions, Pell?s equation, units in real quadratic fields.