Bulletin of the Australian Mathematical Society

Research Article

Hecke groups and continued fractions

David Rosena1 and Thomas A. Schmidta2

a1 Department of Mathematics, Swarthmore College, Swarthmore PA 19081, United States of America

a2 Department of Mathematics, Widener University, Chester PA 19013, United States of America

Abstract

The Hecke groups

S0004972700012120_eqnU1

are Fuchsian groups of the first kind. In an interesting analogy to the use of ordinary continued fractions to study the geodesics of the modular surface, the λ-continued fractions (λF) introduced by the first author can be used to study those on the surfaces determined by the Gq. In this paper we focus on periodic continued fractions, corresponding to closed geodesics, and prove that the period of the λF for periodic S0004972700012120_inline1 has nearly the form of the classical case. From this, we give: (1) a necessary and sufficient condition for S0004972700012120_inline1 to be periodic; (2) examples of elements of xs211Aq) which also have such periodic expansions; (3) a discussion of solutions to Pell's equation in quadratic extensions of the xs211Aq); and (4) Legendre's constant of diophantine approximation for the Gq, that is, γq such that S0004972700012120_inline2 < γq/Q2 implies that P/Q of “reduced finite λF form” is a convergent of real α xs2209 Gq(∞).

(Received November 27 1991)