International Journal of Analysis and Applications

Volume 16, Number 3 (2018), 368-373

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-16-2018-368

SOME REMARKS ON THE ZEROS OF TRIBONACCI POLYNOMIALS

ÖZNUR ÖZTUNÇ KAYMAK∗

Balıkesir University, 10145 Balıkesir, Turkey

∗Corresponding author: oztunc@balikesir.edu.tr

Abstract. In this paper, the zeros of Tribonacci polynomials are studied. The bound containing the zeros

of Tribonacci polynomials has been numerically examined with comparisons. On the other hand, a new

algorithm is given so that it can be used in other boundary problems.

1. Introduction

The features of polynomials have played an important role in many scientific areas such as control theory,

signal processing, crytography and mathematical biology. If we have an accurate estimate of the region con-

taining all the zeros of a polynomial, then the amount of work needed to find exact zeros can be considerably

reduced in comparison with using the classical methods, and for this reason there is always a need for better

and better estimates for the region containing all the zeros of a polynomial. So, there have been a number

of theorems on computations for the roots of polynomials. We consider the generalization of well-known

Fibonacci polynomials called R−Bonacci polynomials. The polynomial Rn(x) is defined by the following

recursive equation in [6] for any integer n and r ≥ 2 :

Rn(x) =

[ (r−1)(n−1)r ]∑
j=0


 n− j − 1

j




r

x(r−1)(n−1)−rj, (1.1)

Received 2017-12-19; accepted 2018-02-12; published 2018-05-02.

2010 Mathematics Subject Classification. 11B39, 12E10, 30C15.

Key words and phrases. Tribonacci polynomials; R-Bonacci polynomials; Fibonacci Numbers.

c©2018 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

368

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-16-2018-368


Int. J. Anal. Appl. 16 (3) (2018) 369

where an =


 n

r




r

is the r−nomial coefficient. Recently, some results have been given on the zeros of

R−Bonacci polynomials Rn(x) when r = 2, 3 in [5]. M. X. He, P. E. Ricci and D. Simon found interesting

curves composing of the zeros of R−Bonacci polynomials in [12]. (If we set r = 2, 3, then the classical

Fibonacci polynomials and Tribonacci polynomials are obtained respectively.) Explicit forms have been

given for the roots Fibonacci polynomials in [7]. Since these forms have been used in many practical areas

eg. Graph Energy, Stable Polynomials, Hand Printed Characters in [15], [16], [18] and [19], it is important

to find the zeros of these polynomials. On the other hand, Moore and Prodinger examined the asymptotic

behaviour of the maximal roots of Fibonacci-type polynomials in [2] and [9], respectively. Furthermore,

the absolute values of complex zeros of Fibonacci-like polynomials have been investigated by Matyas in [8].

Unlike Fibonacci polynomials, explicit forms for the zeros of Tribonacci polynomials have not been found

yet. Instead, zero attractors of these polynomials for r = 3 are determined by W. Goh, M. X. He and P. E.

Ricci in [11]. The symmetric polynomials of the zeros of Tribonacci polynomials are found by M. X. He,

D. Simon and P. E. Ricci in [3]. These results are generalized for the zeros of R−Bonacci polynomials and

their derivatives in [17].

The aim of this paper is to determine the desired region on the complex plane containing the zeros

of Tribonacci polynomials. To support this region, we present numerical results for these polynomials to

compare the results with the known regions. Afterwards, we develop a new algorithm to use in numerical

calculations.

2. Bounds For Polynomials

There have been a number of theorems on computations for the roots of polynomials. These studies date

back to the work of Cauchy [1]. Let

P(z) = zn +

n−1∑
k=0

ak z
k, ak ∈ C (2.1)

be a complex polynomial of degree n. All the zeros of the polynomial P(z) lie in the region

B = {z : |z| < 1 + γ} , (2.2)

where γ = max0≤k≤n−1 |ak|.

Recently, Diaz-Barrero have improved this region by describing two annuli containing all the zeros of

a polynomial where the inner and outer radii are stated in terms of the well-known Fibonacci numbers,

respectively [13].



Int. J. Anal. Appl. 16 (3) (2018) 370

Theorem 2.1. [13] Let P(z) =
n∑

k=0

ak z
k (ak 6= 0, 1 ≤ k ≤ n) be a non-consant polynomial with complex

coefficients. Then, all its zeros lie in the ring shaped region

<1 = {z ∈ C : r1 ≤ |z| ≤ r2} , (2.3)

where

r1 =
3

2
min

1≤k≤n

{
2nFk C(n,k)

F4n

∣∣∣∣a0ak
∣∣∣∣
}1

k

(2.4)

and

r2 =
2

3
max

1≤k≤n

{
F4n

2nFk C(n,k)

∣∣∣∣an−kan
∣∣∣∣
}1

k

. (2.5)

For another annulus, they considered Lucas numbers which are defined by Ln = Ln−1+Ln−2 for n ≥ 0 with

the initial conditions L0 = 2, L1 = 1 (see [5] and [10] for the basic properties of these number sequences).

Theorem 2.2. [4] Let P(z) =
n∑

k=0

ak z
k (ak 6= 0, 0 ≤ k ≤ n) be a non-consant polynomial with complex

coefficients. Then its all zeros of P(z) lie in the annulus

<2 = {z ∈ C : r1 ≤ |z| ≤ r2} , (2.6)

where

r1 = min
1≤k≤n

{
Lk

Ln+2 − 3

∣∣∣∣a0ak
∣∣∣∣
}1

k

(2.7)

and

r2 = max
1≤k≤n

{
Ln+2 − 3

Lk

∣∣∣∣an−kan
∣∣∣∣
}1

k

. (2.8)

3. Tribonacci polynomials

For r = 3 in (1.1), R−Bonacci polynomials are named Tribonacci polynomials, defined by the recurrence

relation

Tn+3 (x) = x
2Tn+2 (x) + xTn+1 (x) + Tn (x) (3.1)

with T0 (x) = 0, T1 (x) = 1 and T2 (x) = x
2. Despite its complicate form, the coefficients of these polynomials

have many interesting properties. It is known that Tribonacci zeros constitute 3-stars as shown below in

Figure 1.

As seen in the above Figure 1, the zeros of these polynomial being invariant with rotation 2π/3 are

separated into 3 sets. So, it may be interesting to find a smallest disc that contains the zeros of these

polynomials since the explicit expressions for the zeros of these polynomials are not known. In order to

examine a new disc containing the zeros of a given polynomial in (3.1), we have used the following identity

n∑
i=1

Fi = Fn+2 − 1, (3.2)



Int. J. Anal. Appl. 16 (3) (2018) 371

Figure 1. The Zeros of T25(x)

where Fn is the n−th Fibonacci number in [14]. Recall that Fibonacci numbers are defined by Fn =

Fn−1 + Fn−2 for n ≥ 2 with the initial conditions F0 = 0, F1 = 1. This identity (3.2) can be found in [5].

Now, we consider the following theorem.

Theorem 3.1. [14] Let P(z) =
n∑

k=0

ak z
k (ak 6= 0, 0 ≤ k ≤ n) be a non-consant complex polynomial. Then

all its zeros lie in the annulus

<3 = {z ∈ C : r1 ≤ |z| ≤ r2} , (3.3)

where

r1 = min
1≤k≤n

{
Fk

Fn+2 − 1

∣∣∣∣a0ak
∣∣∣∣
}1

k

(3.4)

and

r2 = max
1≤k≤n

{
Fn+2 − 1

Fk

∣∣∣∣an−kan
∣∣∣∣
}1

k

. (3.5)

Now we give the following theorem.

Theorem 3.2. Let Tn(x) be a Tribonacci polynomial with n ≡ k(mod 3). Then all its zeros lie

<3 =

{
z ∈ C : r1 ≤ |z| ≤

{
Fn+2 − 1

F3
|an−3|

}1
3

}
, (3.6)

where r1 =
{

F3
Fn+2−1

∣∣∣ 1a3 ∣∣∣}13 for k = 1 and r1 = 0 for k = 0, 2.
Proof. The result is obviously obtained when the r-nomial coefficients are written instead �



Int. J. Anal. Appl. 16 (3) (2018) 372

It will be immediately noticed that the areas containing the zeros of Tribonacci polynomials will be a disc

or an annulus. After reaching this result for the zeros of these polynomials, it is numerically examined by

comparing with the existing boundaries. When comparisons are made, one of the oldest boundaries and the

latest bounds are selected. Below is a table giving inner, outer radius and the area for the zeros of these

polynomials.

Polynomials <1 − Area <2 − Area Cauchy′s Bound <3 − Area

T3 (x) |z| ≤ 1.31 − 5.39 |z| ≤ 1.55 − 7.54 |z| ≤ 2 − 12.56 |z| ≤ 1.51 − 7.16

T4 (x) 0.57 ≤ |z| ≤ 2.20 − 14.18 0.35 ≤ |z| ≤ 2.80 − 24.24 |z| ≤ 3 − 28.27 0.36 ≤ |z| ≤ 2.71 − 22.66

T5 (x) |z| ≤ 4.07 − 52.04 |z| ≤ 4.48 − 63.05 |z| ≤ 4 − 50.26 |z| ≤ 4.32 − 58.62

T6 (x) |z| ≤ 7.90 − 196.06 |z| ≤ 6.83 / 146.55 |z| ≤ 7 − 153.93 |z| ≤ 6.58 − 136.02

T7 (x) 0.10 ≤ |z| ≤ 15.81 − 785.22 0.09 ≤ |z| ≤ 10.16 − 324.26 |z| ≤ 11 − 380.13 0.10 ≤ |z| ≤ 9.79 − 301.07

T8 (x) |z| ≤ 32.28 − 3273.53 |z| ≤ 14.89 − 696.52 |z| ≤ 17 − 907.92 |z| ≤ 14.35 − 646.92

T9 (x) |z| ≤ 66.92 − 14069 |z| ≤ 21.62 − 1468.46 |z| ≤ 31 − 3019.07 |z| ≤ 20.83 − 1363.1

T10 (x) 0.14 ≤ |z| ≤ 140.30 − 61839.3 0.03 ≤ |z| ≤ 31.15 − 3048.36 |z| ≤ 51 − 8171.28 0.03 ≤ |z| ≤ 30.02 − 2831.2

Above, the upper bounds containing the zeros of the first ten Tribonacci polynomials are shown numer-

ically. (Since the zero of T0 (x) ,T2 (x) is only 0 and T1 (x) is a fixed polynomial, they are not included in

this table.) At first glance, it seems that Diaz-Barrero’s boundary(<1) is the most useful among others.

But, it is clearly observed that actual area is gradually growing except for first few Tribonacci polynomials.

And this shows that this bound is unusable for these polynomials. On the other hand, it is clear that our

boundary(<3) gives better results in terms of both r1, r2 and area, when compared to the bound of Cauchy,

one of the oldest boundaries and the bound of Dalal and Govil(<2) which have been recently obtained.

Although not included in the table, these calculations have been numerically verified for the first 20 Tri-

bonacci polynomials. Hence, the disk or annulus obtained by our boundary and containing the zeros of each

Tribonacci polynomial will remain the smallest. This is the desired ideal situation.

4. Application

In this section, we develop an algorithm. At the beginning of the algorithm, all Tribonacci polynomials are

generated and then the coefficients are taken to be used at any boundary. The algorithm makes it easy for

us to make this computation for the Tribonacci Polynomial as many as we would like to. Furthermore, this

algorithm can be developed and adapted for other boundary value problems. And, thanks to this algorithm,

testing of the usability of any boundary can be done without needing long and complicated operations.



Int. J. Anal. Appl. 16 (3) (2018) 373

Table 1. The Algorithm

t[0]=0;

t[1]=1;

t[2] = x2;

For[i=2,i¡16,i++,

t[n](x ):=t(i+1)=x2t(i) + xt(i− 1) + t(i− 2);

a[n]:=Coefficient[t(n)(x),x,2 i];

a[n-3]:=Coefficient[t(n)(x),x,2 i-3];

s[n]:=Coefficient[t(n)(x),x,0];

e[n]:=Exponent[t(n)(x),x];

Print

[
max

({
N

[
3

√
|a(n−3)a(n) |(Fe(n)+2−1)

F3

]})]]

References

[1] A. L. Cauchy, Exercises de Mathématiques IV, Anné de Bure Fréres, Paris, 1829.

[2] G. Moore, The limit of the golden numbers is 3/2, Fibonacci Quart. 32 (1994), 211-217.

[3] M. X. He, D. Simon, P. E. Ricci, Dynamics of the zeros of Fibonacci polynomials, Fibonacci Quart. 35 (1997), 160-168.

[4] A. Dalal, N. K. Govil, On Region Containing all the Zeros of a Polynomial, Appl. Math. Comput. 219 (2013), 9609-9614.

[5] T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, 2001.

[6] V. E. Hoggat, M. Bicknell, Generalized Fibonacci polynomials, Fibonacci Quart. 11 (1973), 457-465.

[7] V. E. Hoggat, M. Bicknell, Roots of Fibonacci polynomials, Fibonacci Quart. 11 (1973), 271-274.

[8] F. Matyas, Bounds for the zeros of Fibonacci-Like Polynomials, Acta Acad. Paed. Agriensis, Sec. Math. 25 (1998), 15-20.

[9] H. Prodinger, The asymptotic behavior of the golden numbers, Fibonacci Quart. 34 (1996), 224-225.

[10] J. H. Conway, R. Guy, The Book of Numbers, Copernicus, New York, 1996.

[11] W. Goh, M. X. He, P. E. Ricci, On the universal zero attractor of the Tribonacci-related polynomials, Calcolo 46 (2009),

95-129.

[12] M. X. He, P. E. Ricci, D. Simon, Numerical results on the zeros of generalized Fibonacci polynomials, Calcolo 34 (1997),

25-40.

[13] J.L. Diaz-Barrero, An annulus for the zeros of polynomials, J. Math. Anal. Appl. 273 (2002), 349-352.

[14] Ö. Öztunç, R−Bonacci Polynomials and Their Derivatives, Ph. D. Thesis, Balıkesir University, (2014).

[15] X. Li, Y. Shi, I. Gutman, Graph Energy, Springer, New York, 2012.

[16] S. Fisk, Polynomials, roots, and interlacing, ArXiv mat/0612833v2, (2006).

[17] N. Y. Özgür, Ö. Öztunç, On the Zeros of R−Bonaci Polynomials, ArXiv:1711.01150, (2017).

[18] P. G. Anderson, R. S. Gaborski, The Polynomial Method Augmented by Supervised Training for Hand-Printed Character

Recognition, Artificial Neural Networks and Genetic Algorithms, Proceedings of the International Conference, (1993).

[19] S. Ucar, N. Y. Ozgur, Right Circulant Matrices with Generalized Fibonacci and Lucas Polynomials and Coding Theory,

ArXiv:1801.01766 [math.CO].


	1. Introduction
	2. Bounds For Polynomials
	3. Tribonacci polynomials
	4. Application
	References