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Cyclotomy-generated polynomials of Fibonacci type. (English) Zbl 0593.12007
Fibonacci numbers and their applications, Pap. 1st Int. Conf., Patras/Greece 1984, Math. Appl., D. Reidel Publ. Co. 28, 81-97 (1986).
The ”cyclotomic” polynomial \(\Phi_n(x)\) \((=\Phi_n\) for convenience) is defined as
\[ \Phi_n=(x^ n-1)/(x-1)\quad\text{ if }x\neq 1 \quad \Phi_n =n\quad\text{ if }x=1, \quad (\Phi_0=0). \]
The cyclotomy-generated polynomial of Fibonacci type of degree \(n^2+n\) is
\[ \Phi_{n^2+n}=x^{n^2+n}- x^{2n+1}\frac{\Phi_{n^2- 1}}{\Phi_{n+1}}+x^{2n}\frac{\Phi_{n^2-n}}{\Phi_n}. \]
Various theorems are proved, and conjectures suggested, about the roots of this polynomial equation, written as \(p_n(x)=0\). Graphs of \(y=p_n(x)\) for \(n=1, 2, 3, 4\) are drawn and their common features indicated.
[For the entire collection see Zbl 0582.00009.]
Reviewer: Alwyn F. Horadam

MSC:
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11C08 Polynomials in number theory