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
Tc:=(c2+c+1−cc21−c)∈Γ0(c2), c∈Z, c≠0,Tc:=(c2+c+1−cc21−c)∈Γ0(c2), c∈Z, c≠0,we determined the orbit
{Trc(∞):r∈N}.{Tcr(∞):r∈N}. Each rational number of this set is the form Pr(c)/Qr(c),Pr(c)/Qr(c), where Pr(c)Pr(c) and Qr(c)Qr(c) are the polynomials in Z[c]Z[c]. It is shown that Pr(1)Pr(1) and Qr(1)Qr(1) the sum of the coefficients of the polynomials Pr(c)Pr(c) and Qr(c)Qr(c) 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
Qr(c)=r∑s=1(2r−ss−1)c2r−2s+2Qr(c)=∑s=1r(2r−ss−1)c2r−2s+2
Birincil Dil | İngilizce |
---|---|
Konular | Matematik |
Bölüm | Research Article |
Yazarlar | |
Yayımlanma Tarihi | 30 Mart 2022 |
Gönderilme Tarihi | 18 Mayıs 2021 |
Kabul Tarihi | 22 Ekim 2021 |
Yayımlandığı Sayı | Yıl 2022 Cilt: 71 Sayı: 1 |
Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.
This work is licensed under a Creative Commons Attribution 4.0 International License.