We consider the extended Hecke groups H (q / generated by T .´/ D 1 =´, S.´/ D 1=.´C q / and R.´/ D 1=´with q D 2 cos. =q/ for q 3 integer: In this article, we study the abstract group structures of the power subgroups H m (q / of H (q / for each positive integer m. Then, we give the relations between commutator subgroups and power subgroups.

In this paper, we define a sequence, which is a generalized version of the Lucas sequence, similar to the generalized Fibonacci sequence given in Koruoglu andŞahin in Turk. J. Math. 2009, doi:10.3906/mat-0902-33. Also, we give some connections between the generalized Fibonacci sequence and the generalized Lucas sequence, and we find polynomial representations of the generalized Fibonacci and the generalized Lucas sequences, related to the extended Hecke groups given in