Generalized fibonacci sequences related to the extended hecke groups and an application to the extended modular group

Abstract

The extended Hecke groups (H) over bar (lambda(q)) are generated by T(z) = -1/z, S(z) = 1/(z + lambda(q)) and R(z) = 1/(z) over bar with lambda(q) = 2 cos(pi/q) for q >= 3 integer. In this paper, we obtain a sequence which is a generalized version of the Fibonacci sequence given in [6] for the extended modular group (Gamma) over bar, in the extended Hecke groups (H) over bar(lambda(q)). Then we apply our results to (Gamma) over bar to find all elements of the extended modular group (Gamma) over bar.