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

Authors: ÖZDEN KORUOĞLU, RECEP ŞAHİN

Abstract: The extended Hecke groups \overline{H}(\lambda _{q}) are generated by T(z)=-1/z, S(z)=-1/(z+\lambda _{q}) and R(z)=1/ \overline{z} with \lambda _{q}=2\cos (\pi /q) for q\geq 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 \overline{H}(\lambda_{q}). Then we apply our results to \overline{\Gamma } to find all elements of the extended modular group \overline{\Gamma }.

Keywords: Extended Hecke groups, extended modular group, Fibonacci numbers

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