Limits of zeros of polynomial sequences.(English)Zbl 1171.11009
The \(k\)-th order Fibonacci sequence is defined by the recurrence \[G(k,l)=G(k,l-1)+G(k,l-2)+\dots+G(k,l-k)\] with the initial value \(G(k,1)=1\). One may see that, for example, \(G(2,l)=G(2,l-1)+G(2,l-2)=F_l,\) where \(F_l\) is the \(l\)-th Fibonacci number. The characteristic polynomials for these sequences are defined by \(F_k(x)=x^k-x^{k-1}-x^{k-2}-\dots-1.\)
There are some conjectures of G. Grossman and S. Narayan concerning to the zeros of the polynomials \(F_k(x)\) and their derivatives. The authors anwer these in the affirmative. To obtain the results, they take the polynomials \(D_j(x)=F_{j+l}^{(l)}(x)\) (\(l\) is fixed).
In what follows, \(u_j\) denotes the positive root of \(D_j(x)\) and – if \(j\) is even – \(v_j\) denotes the negative root of \(D_j(x)\). (That really one or two real zeros exist, comes from the theorem below.)
Theorem 2.1: (1) Let \(j=k-l\). Then \(\lim_{j\rightarrow\infty}u_j=2.\) All of the other complex roots of \(D_j(x)\) are inside of \(|z|<u_j\). For even \(j\), we have \(\lim_{j\rightarrow\infty}v_j=-1.\)
(2) If \(j\) is odd, then \(D_j(x)\) has one positive root and no negative roots. If \(j\) is even, then \(D_j(x)\) has one positive and one negative roots.
(3) For \(j\geq 2\) we have \(u_{j+1}>u_j\).
(4) There exists a number \(N_0\), such that for even \(n>N_0\), we have \(v_{n+2}<v_n\).
The authors prove the corresponding result for the integrals of the characteristic polynomials, too.
There are some conjectures of G. Grossman and S. Narayan concerning to the zeros of the polynomials \(F_k(x)\) and their derivatives. The authors anwer these in the affirmative. To obtain the results, they take the polynomials \(D_j(x)=F_{j+l}^{(l)}(x)\) (\(l\) is fixed).
In what follows, \(u_j\) denotes the positive root of \(D_j(x)\) and – if \(j\) is even – \(v_j\) denotes the negative root of \(D_j(x)\). (That really one or two real zeros exist, comes from the theorem below.)
Theorem 2.1: (1) Let \(j=k-l\). Then \(\lim_{j\rightarrow\infty}u_j=2.\) All of the other complex roots of \(D_j(x)\) are inside of \(|z|<u_j\). For even \(j\), we have \(\lim_{j\rightarrow\infty}v_j=-1.\)
(2) If \(j\) is odd, then \(D_j(x)\) has one positive root and no negative roots. If \(j\) is even, then \(D_j(x)\) has one positive and one negative roots.
(3) For \(j\geq 2\) we have \(u_{j+1}>u_j\).
(4) There exists a number \(N_0\), such that for even \(n>N_0\), we have \(v_{n+2}<v_n\).
The authors prove the corresponding result for the integrals of the characteristic polynomials, too.
Reviewer: István Mező (Debrecen)
MSC:
| 11B39 | Fibonacci and Lucas numbers and polynomials and generalizations |
