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

The extended Hecke groups H(λq) are generated by T(z) = −1/z , S(z) = −1/(z +λq) and R(z) = 1/ z with λq = 2cos(π/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 overline{Gamma }, in the extended Hecke groups H(λq) . Then we apply our results to overline{Gamma } to find all elements of the extended modular group overline{Gamma }.