AI Summary & Key Takeaways
AI InsightIt is shown that P r ( 1 ) Pr(1) and Q r ( 1 ) Qr(1) the sum of the coefficients of the polynomials P r ( c ) Pr(c) and Q r ( 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 Q r ( c ) = r ∑ s = 1 ( 2 r − s s − 1 ) c 2 r − 2 s + 2 Qr(c)=∑s=1r(2r−ss−1)c2r−2s+2
📌 3 Key Takeaways
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 : = ( c 2 + c + 1 − c c 2 1 − c ) ∈ Γ 0 ( c 2 ) , c ∈ Z , c ≠ 0 , Tc:=(c2+c+1−cc21−c)∈Γ0(c2), c∈Z, c≠0, we determined the orbit { T r c ( ∞ ) : r ∈ N } . {Tcr(∞):r∈N}. Each rational number of this set is the form P r ( c ) / Q r ( c ) , Pr(c)/Qr(c), where P r ( c ) Pr(c) and Q r ( c ) Qr(c) are the polynomials in Z [ c ] Z[c] . It is shown that P r ( 1 ) Pr(1) and Q r ( 1 ) Qr(1) the sum of the coefficients of the polynomials P r ( c ) Pr(c) and Q r ( 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 Q r ( c ) = r ∑ s = 1 ( 2 r − s s − 1 ) c 2 r − 2 s + 2 Qr(c)=∑s=1r(2r−ss−1)c2r−2s+2