Some group actions and Fibonacci numbers

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

$T_c:=\left(\begin{array}{cc}{c}^2+c+1& -c\\ c^2& 1-c\end{array}\right)\in {\Gamma }_0\left(c^2\right),c\in Z,c\ne 0,$

we determined the orbit 

$\left\{T_c^r\right(\infty ):r\in N\}.$ Each rational number of this set is the form $P_r\left(c\right)/Q_r\left(c\right),$ where $P_r\left(c\right)$ and $Q_r\left(c\right)$ are the polynomials in $Z\left[c\right]$. It is shown that $P_r\left(1\right)$ and $Q_r\left(1\right)$ the sum of the coefficients of the polynomials $P_r\left(c\right)$ and $Q_r\left(c\right)$ respectively, are the Fibonacci numbers, where

$P_{r}(c)=\sum \limits_{s=0}^{r}(
\begin{array}{c}
2r-s \\
s
\end{array}
) c^{2r-2s}+\sum \limits_{s=1}^{r}(
\begin{array}{c}
2r-s \\
s-1
\end{array}) c^{2r-2s+1}$

and

$Q_r\left(c\right)=\sum _{s=1}^r\left(\begin{array}{c}2r-s\\ s-1\end{array}\right)c^{2r-2s+2}$

Keywords

• Suborbital graphs
• Pascal triangle
• Fibonacci numbers

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