Some group actions and Fibonacci numbers

Authors : Zeynep ŞANLI, Tuncay KÖROĞLU

Pages : 273-284

Doi:10.31801/cfsuasmas.939096

View : 13 | Download : 21

Publication Date : 2022-03-30

Article Type : Research Paper

Abstract :The Fibonacci sequence has many interesting properties and studied by many mathematicians. The terms of this sequence appear in nature and is connected with combinatorics and other branches of mathematics. In this paper, we investigate the orbit of a special subgroup of the modular group. Taking T c : = insert ignore into journalissuearticles values( c 2 + c + 1 − c c 2 1 − c ); ∈ Γ 0 insert ignore into journalissuearticles values( c 2 ); ,   c ∈ Z ,   c ≠ 0 , Tc:=insert ignore into journalissuearticles values(c2+c+1−cc21−c);∈Γ0insert ignore into journalissuearticles values(c2);, c∈Z, c≠0, we determined the orbit  { T r c insert ignore into journalissuearticles values( ∞ ); : r ∈ N } . {Tcrinsert ignore into journalissuearticles values(∞);:r∈N}. Each rational number of this set is the form  P r insert ignore into journalissuearticles values( c ); / Q r insert ignore into journalissuearticles values( c ); , Prinsert ignore into journalissuearticles values(c);/Qrinsert ignore into journalissuearticles values(c);, where P r insert ignore into journalissuearticles values( c ); Prinsert ignore into journalissuearticles values(c); and Q r insert ignore into journalissuearticles values( c ); Qrinsert ignore into journalissuearticles values(c); are the polynomials in Z [ c ] Z[c] . It is shown that  P r insert ignore into journalissuearticles values( 1 ); Prinsert ignore into journalissuearticles values(1); and Q r insert ignore into journalissuearticles values( 1 ); Qrinsert ignore into journalissuearticles values(1); the sum of the coefficients of the polynomials  P r insert ignore into journalissuearticles values( c ); Prinsert ignore into journalissuearticles values(c); and Q r insert ignore into journalissuearticles values( c ); Qrinsert ignore into journalissuearticles values(c); respectively, are the Fibonacci numbers, where $P_{r}insert ignore into journalissuearticles values(c);=\sum \limits_{s=0}^{r}insert ignore into journalissuearticles values( \begin{array}{c} 2r-s \\ s \end{array} ); c^{2r-2s}+\sum \limits_{s=1}^{r}insert ignore into journalissuearticles values( \begin{array}{c} 2r-s \\ s-1 \end{array}); c^{2r-2s+1}$ and Q r insert ignore into journalissuearticles values( c ); = r ∑ s = 1 insert ignore into journalissuearticles values( 2 r − s s − 1 ); c 2 r − 2 s + 2 Qrinsert ignore into journalissuearticles values(c);=∑s=1rinsert ignore into journalissuearticles values(2r−ss−1);c2r−2s+2
Keywords : Suborbital graphs, Pascal triangle, Fibonacci numbers