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Further generalization of the Fibonacci-coefficient polynomials
Abstract
The aim of this paper is to investigate the zeros of the general polynomials q(i,t) n (x) = n X k=0 Ri+ktx n k = Rix n + Ri+tx n 1 + ··· + Ri+(n 1)tx + Ri+nt, where i > 1 and t > 1 are fixed integers.
... Garth, Mills, and Mitchell [1] considered the Fibonacci coefficient polynomial defined by p n (x) = F 1 x n + F 2 x n−1 + · · · + F n x + F n+1 and showed that it has no real zeros if n is even and exactly one real zero if n is odd. This result was later extended by Mátyás [4,5] to polynomials whose coefficients are given by more general second order recurrences (having constant coefficients). Mátyás and Szalay [6] showed that the same also holds true for the Tribonacci coefficient polynomials q n (x) = T 2 x n + T 3 x n−1 + · · · + T n+1 x + T n+2 , a result which has been extended to k-Fibonacci [2] and generalized Tribonacci [3] coefficient polynomials. ...
... To show (5), note that n(n + 2)a 2 n > (n + 1) 2 a 2 n − qa n a n+1 since qa n a n+1 > a 2 n . Thus, by the induction hypothesis, we have n(n + 2)a n−1 a n+1 > n(n + 2)a 2 n > (n + 1) 2 a 2 n − qa n a n+1 , which gives (5) and completes the proof. ...
Let denote the distribution on the set of involutions of size n for the statistic which records the number of fixed points. We show for a range of q values that the polynomial always has the smallest possible number of real zeros, that is, none when the degree is even and one when the degree is odd. On the way, we show that the sequence is log-convex for all . Our proof in the case q=1 is elementary, while the proof for the general case makes use of programming to estimate the zeros of some related analytic functions in q. Furthermore, the sequence of real zeros obtained from the odd case is shown to be monotonically convergent. We also consider the polynomial , where denotes the number of derangements of an i-element set, and show that the same holds for it. Our results extend recent ones concerning Fibonacci and Tribonacci coefficient polynomials.
... D. Garth, D. Mills and P. Mitchell [1] introduced the definition of the Fibonaccicoefficient polynomials p n (x) = F 1 x n + F 2 x n−1 + · · · + F n x + F n+1 and -among others -determined the number of the real zeros of p n (x). In [2] we investigated the zeros of the much more general polynomials ...
The aim of this paper is to investigate the zeros of polynomials P n,k (x)=K k-1 x n +K k x n-1 +⋯+K n+k2 x+K n+k-1 , where the coefficients K i are terms of a linear recursive sequence of k-order (k≥2).
... Garth, Mills, and Mitchell [3] considered the Fibonacci coefficient polynomial p n (x) = F 1 x n + F 2 x n−1 + · · · + F n x + F n+1 and showed that it has no real zeros if n is even and exactly one real zero if n is odd. Later, Mátyás [5,6] extended this result to polynomials whose coefficients are given by more general second order recurrences (having constant coefficients), and Mátyás and Szalay [7] showed the same holds true for the Tribonacci coefficient polynomials q n (x) = T 2 x n + T 3 x n−1 + · · · + T n+1 x + T n+2 . The latter result has been extended to k-Fibonacci polynomials by Mansour and Shattuck [4]. ...
We consider two general classes of second-order linear recurrent sequences and the polynomials whose coefficients belong to a sequence in either of these classes. We show for each such sequence {ai} i≥0 that the polynomial f (x) = n i=0 aix i always has the smallest possible number of real zeros, that is, none when the degree is even and one when the degree is odd. Among the sequences then for which this is true are the Motzkin, Riordan, Schröder, and Delannoy numbers.
... In particular, they showed that p n (x) has no real zeros if n is even and exactly one real zero if n is odd. Later, this result was extended by Mátyás [5,6] to more general second order recurrences. Mátyás and Szalay [8] showed that the same result also holds for the Tribonacci coefficient polynomials q n (x) = T 2 x n + T 3 x n−1 + · · · + T n+1 x + T n+2 . ...
Let denote the third order linear recursive sequence defined by the initial values and and the recursion if , where p, q, and r are constants. The are \emph{generalized Tribonacci} numbers and reduce to the usual \emph{Tribonacci} numbers when p=q=r=1 and to the \emph{3-bonacci} numbers when p=r=1 and q=0. Let , which we'll refer to as a \emph{generalized Tribonacci coefficient polynomial}. In this paper, we show that the polynomial has no real zeros if n is even and exactly one real zero if n is odd, under the assumption that p and q are non-negative real numbers with . This generalizes the known result when p=q=r=1 and seems to be new in the case when p=r=1 and q=0. Our argument when specialized to the former case provides an alternative proof of that result. We also show, under the same assumptions for p and q, that the sequence of real zeros of the polynomials when n is odd converges to the opposite of the positive zero of the characteristic polynomial associated with the sequence . In the case p=q=r=1, this convergence is monotonic. Finally, we are able to show the convergence in modulus of all the zeros of when .
... In particular, they showed that p n (x) has no real zeros if n is even and exactly one real zero if n is odd. Later, this result was extended by Mátyás [5, 6] to more general second order recurrences . The same result also holds for the Tribonacci coefficient polynomials ...
Let denote the linear recursive sequence of order k () defined by the initial values and and the recursion if . The are often called \emph{k-Fibonacci} numbers and reduce to the usual \emph{Fibonacci} numbers when k=2. Let , which we will refer to as a \emph{k-Fibonacci coefficient} polynomial. In this paper, we show for all k that the polynomial has no real zeros if n is even and exactly one real zero if n is odd. This generalizes the known result for the k=2 and k=3 cases corresponding to Fibonacci and Tribonacci coefficient polynomials, respectively. It also improves upon a previous upper bound of approximately k for the number of real zeros of . Finally, we show for all k that the sequence of real zeros of the polynomials when n is odd converges to the opposite of the positive zero of the characteristic polynomial associated with the sequence . This generalizes a previous result for the case k=2.
This paper shows, that the Tribonacci-coefficient polynomial P n (x)=T 2 x n +T 3 x n-1 +⋯+T n+1 x+T n+2 has exactly one real zero if n is odd, and P n (x) does not vanish otherwise. This improves the result in F. Mátyás, K. Liptai, J. T. Tóth and F. Filip [Ann. Math. Inform. 37, 101–106 (2010; Zbl 1224.11046)], which provides the upper bound 3 or 2 on the number of zeros of P n (x), respectively.
The Fibonacci sequence's initial terms are F_0=0 and F_1=1, with F_n=F_{n-1}+F_{n-2} for n>=2. We define the polynomial sequence p by setting p_0(x)=1 and p_{n}(x)=x*p_{n-1}(x)+F_{n+1} for n>=1, with p_{n}(x)= sum_{k=0}^{n} F_{k+1}x^{n-k}. We call p_n(x) the Fibonacci-coefficient polynomial (FCP) of order n. The FCP sequence is distinct from the well-known Fibonacci polynomial sequence. We answer several questions regarding these polynomials. Specifically, we show that each even-degree FCP has no real zeros, while each odd-degree FCP has a unique, and (for degree at least 3) irrational, real zero. Further, we show that this sequence of unique real zeros converges monotonically to the negative of the golden ratio. Using Rouche's theorem, we prove that the zeros of the FCP's approach the golden ratio in modulus. We also prove a general result that gives the Mahler measures of an infinite subsequence of the FCP sequence whose coefficients are reduced modulo an integer m>=2. We then apply this to the case that m=L_n, the nth Lucas number, showing that the Mahler measure of the subsequence is phi^{n-1}, where phi=(1+sqrt 5)/2.
In this note we deal with the zeros of polynomials defined recursively, where the coefficients of these polynomials are the terms of a given second order linear recursive sequence of integers. Some results on the Fibonacci-coefficient polynomials obtained by D. Garth, D. Mills and P. Mitchell will be generalized.