Abstract
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 and Sahin 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 Koruoglu and Sahin in Turk. J. Math. 2009, doi:10.3906/mat-0902-33.
Highlights
In [ ], Hecke introduced groups H(λ), generated by two linear fractional transformationsT(z) = – and S(z) =, z z+λ where λ is a fixed positive real number
We define a sequence, which is a generalized version of the Lucas sequence, similar to the generalized Fibonacci sequence given in Koruoglu and Sahin in Turk
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 Koruoglu and Sahin in Turk
Summary
We define a sequence, which is a generalized version of the Lucas sequence, similar to the generalized Fibonacci sequence given in Koruoglu and Sahin in Turk. The extended Hecke group H(λq) has a presentation We can represent the generators of the extended Hecke group H(λq) as and R = In [ ], Koruoglu and Sahin found that there is a relationship between the generalized Fibonacci numbers and the entries of matrices representations of some elements of the extended Hecke group H(λq). Notice that this real numbers sequence is a generalized version of the common Fibonacci sequence.
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