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ON THE ZEROS OF R -BONACCI POLYNOMIALS AND THEIR DERIVATIVES

. The purpose of the present paper is to examine the zeros of R -Bonacci polynomials and their derivatives. We conﬁrm a conjecture about the zeros of R -Bonacci polynomials for some special cases. We also ﬁnd explicit formulas of the roots of derivatives of R -Bonacci polynomials in some special cases.


Introduction
The problem finding a convenient method to determine the zeros of a polynomial has a long history that dates back to the work of Cauchy [9]. In this paper our aim is to examine the zeros of R-Bonacci polynomials and their derivatives. R-Bonacci polynomials R n (x) are defined by the following recursive equation in [6] for any integer n and r ≥ 2 : with the initial values R −k (x) = 0, k = 0, 1, ..., r − 2, R 1 (x) = 1, R 2 (x) = x r−1 . For r = 2 we obtain the classical Fibonacci polynomials. There are a great number of publications regarding to Fibonacci polynomials and their generalizations, (see [4]- [6], [8] and [11]). In [7], V. E. Hoggat and M. Bicknell found the zeros of these polynomials using hyperbolic trigonometric functions. For r = 3 we obtain Tribonacci polynomials. Since the open expressions are not found for the zeros of Tribonacci polynomials and their derivatives, numerical studies have been studied in recent years. Zero attractors of these polynomials are obtained by W. Goh, M. X. He and P. E. Ricci in [3]. In [10], the number of the real roots of Tribonacci-coefficient polynomials are found. The symmetric polynomials of the zeros of Fibonacci and Tribonacci are found by M. X. He, D. Simon and P. E. Ricci in [4]. Furthermore, in [5], it was determined the location and distribution of the zeros of the Fibonacci and Tribonacci polynomials. They found out interesting geometric properties of these polynomials. They proved the following equation r , k = 0, 1, ..., r − 1, n = 1, 2, ... (1.2) and deduce that the zeros of R-Bonacci polynomials lie on the equally spaced r-stars with respect to the argument 2π r . Also they conjectured that these r-stars have r − 1 branches starting at the zeros of the equation x r + 1 = 0.
In Section 2 we confirm this conjecture for some classes of R-Bonacci polynomials. To do this we find the symmetric polynomials which are made up of the r th order of the zeros of R-Bonacci polynomials. Using these symmetric polynomials, we determine the reference roots for the polynomials R rn+p (x) for p = 0, 1 and n = 1.
There have been several papers on the derivatives of the Fibonacci polynomials (see [1], [2] and [12]). In Section 3 we study the roots of the derivatives of R-Bonacci polynomials. We obtain the most general symmetric polynomials which are made up of the r th order of the zeros of derivatives of R-Bonacci polynomials. Using these symmetric polynomials, we find some formulas for the zeros of derivatives of R-Bonacci polynomials for some values of t. These formulas are substantially simple and useful.

Zeros of Some Classes of R-Bonacci Polynomials
It was given the general representations for R-Bonacci polynomials as [6] R n (x) = Here n j r denotes the r-nomial coefficient. In this section, we obtain the symmetric polynomials including the zeros of R-Bonacci polynomials. Our results are coincide with the ones obtained in [4] for R = 2, 3.
For the definition of a symmetric polynomial see [4]. Let {x 1 , ..., x r } be the set of the reference zeros of the polynomial R rn (x).
Theorem 2.1. The most general form of the j th symmetric polynomials consisting of over the r th of zeros of R rn (x) is as follows: Proof. By (1.2), the zeros of R-Bonacci polynomials lie in the argument 2π r and hence the polynomial R rn (x) can be factorized as

If we rearrange this equation we obtain
On the other hand by (2.1) we can write Since the equations (2.3) and (2.4) are equal, we obtain the desired result (2.2).
Corollary 2.1. The following equations are satisfied by the zeros of R rn (x) : Proof. By setting j = 1 in the equation (2.2) desired result is obtained.
The most general form of the j th symmetric polynomials consisting of the r th zeros of R rn+1 (x) is as follows : Proof. By a similar way used in the proof of Theorem 2.1 we can write Then we get By putting rn + 1 instead of n in (2.1) we find It follows from the comparison (2.7) and (2.8) it is possible to write the desired result (2.6).
Corollary 2.2. The following equations are satisfied by the zeros of R rn+1 (x) : Proof. If we set j = 1 in the equation (2.6) then we get the equation (2.9). Now using these symmetric polynomials we obtain the reference roots of R rn+p (x) for p = 0, 1. x r respectively. Rearranging the above equations, it can be easily seen that the reference roots of R rn+p (x) as in the equation (2.10). Using (2.10) if we solve the equation x 5 j = −1(1 ≤ j ≤ 5), the reference roots of the polynomial B 6 (x) are found as follows (see Figure 1) :

Zeros of Derivatives of R-Bonacci Polynomials
Before we find the symmetric polynomials which are made up of the r th order of the zeros of the derivatives of R-Bonacci polynomials R (3.1) Now we determine the symmetric polynomials for R then the most general form of the symmetric polynomials consisting of the zeros of R (t) rn+p (x) is as follows: Proof. It can be easily seen that where µ is a coefficient. Then we have rn+p (x) = µ{x r 2 n−rn−(t+(1−p)(r−1)) − x r 2 n−rn−(t+(1−p)(r−1))−r η k=1 x r k + x r 2 n−rn−(t+(1−p)(r−1))−2r j =k (3.8) and Proof. In the equation (3.5), if we put j = η and j = 1 we obtain the desired results, respectively. Let and Then we can give the following theorem. rn+p (x) has r((r − 1)n − t+(1−p)(r−1) r ) roots and these roots are r , (k = 0, 1, ..., r − 1), (3.12) where υ 2 and ψ 2 are defined by the equations (3.10) and (3.11), respectively.
Proof. Since R x r k = x r 1 + x r 2 = ψ 2 . (3.14) Since we know that x r 1 = υ 2 x r 2 it can be easily seen that x 2r 2 − ψ 2 x r 2 + υ 2 = 0. Solving this last equation of the second degree, the roots can be easily found. So the roots of R Since we have Fibonacci and Tribonacci polynomials for r = 2 and r = 3, respectively, we can give the following corollaries.