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# A note on the roots of some special type polynomials

*Journal of Inequalities and Applications*
**volume 2013**, Article number: 466 (2013)

## Abstract

In this study, we investigate the polynomials for n\ge 2 and positive integers *k* and a positive real number *a*, with the initial values {G}_{0}(x)=-a, {G}_{1}(x)=x-a

We give some fundamental properties related to them. Also, we obtain asymptotic results for the roots of polynomials {G}_{n}^{(k)}(x).

**MSC:**11B39, 11B37.

## 1 Introduction

The polynomials defined by Catalan, for n\ge 0, as follows

are called Fibonacci polynomials and denoted by {F}_{n}(x), [1]. The Fibonacci-type polynomials {G}_{n}(x), n\ge 0, are defined by

where {G}_{0}(x) and {G}_{1}(x) are seed polynomials. There are several studies about the properties of zeros of polynomials {G}_{n}(x). However, there are no general formulas for zeros of Fibonacci-type polynomials. In [2, 3], the authors studied the limiting behavior of the maximal real roots of polynomials {G}_{n}(x) with the initial values {G}_{0}(x)=-1, {G}_{1}(x)=x-1. In [4], the authors generalized Moore’s result for these polynomials. They considered the initial values {G}_{0}(x)=a, {G}_{1}(x)=x+b, where *a* and *b* are integer numbers. In [5], the author determined the absolute values of complex zeros of these polynomials. In [6], Ricci studied this problem in the case a=1 and b=1. In [7], Tewodros investigated the convergence of maximal real roots of different Fibonacci-type polynomials given by the following relation:

where *k* is a positive integer number. The initial values of the recursive relation (3) are {G}_{{}_{0}(k)}^{}(x)=-1 and {G}_{{}_{1}(k)}^{}(x)=x-1. In this study, firstly, we investigate some fundamental properties of Fibonacci-type polynomials. We give some combinatorial identities related to equation (3). Then, we investigate the limit of maximal real roots of these polynomials. We notice that Tewodros [7] studied a special case a=1 of the polynomials we investigate.

## 2 Some fundamental properties of polynomials {G}_{n}^{(k)}(x)

In this section, we give some fundamental properties of polynomials {G}_{{}_{n}(k)}^{}(x), for n\ge 0, defined by the recursive formula as follows:

The characteristic equation for (4) is {t}^{2}-{x}^{k}t-1=0 and its roots are

and

Note that \alpha (x)\beta (x)=-1, \alpha (x)+\beta (x)={x}^{k} and \alpha (x)-\beta (x)=\sqrt{{x}^{2k}+4}. For relation (4), the Binet formula is

where

**Proposition 2.1** *For* n\ge 0, *the generating function for polynomials* {G}_{n}^{(k)}(x) *is*

*Proof* Let {H}_{r}^{(k)}(x,t) be the generating function for polynomials {G}_{n+r}^{(k)}(x). So, we write

If we multiply both sides of equation (8) by {x}^{k}t and {t}^{2}, respectively, then we can get

and

The last two equations give us the following equation:

If we use the recurrence relation and simplify it, we write

*i.e.*,

Thus, the proof is completed. □

Let us give the well-known formula, which is called the Cassini-like formula, without proof.

**Proposition 2.2** (Cassini-like)

*For* n\ge 0, *we have*

*where*

*and*

In the following propositions, we give some sums formulas related to polynomials {G}_{n}^{(k)}(x).

**Proposition 2.3** *For* N\ge 0, *we have*

*Proof* Proof of formula (10) follows now immediately from (7). □

**Proposition 2.4** *For* N\ge 0, *we have the following sum formulas*:

*and*

*Proof* From the Binet formula, we can write

where \alpha =\alpha (x), \beta =\beta (x). If we substitute the equations

and

into equation (13), then we can get

If we rearrange the last equation, then we have

By taking aid of the Binet formula, we can write

If we substitute \alpha +\beta ={x}^{k}, \alpha \beta =-1, (1-{\alpha}^{2})(1-{\beta}^{2})=-{x}^{2k} into the last equation, we obtain the following equation:

Thus, the proof is completed. Similarly, the second part of the proposition can be seen. □

## 3 Asymptotic behaviors of the maximal roots for polynomials {G}_{n}^{(k)}(x)

In this section, firstly for k=2, we investigate the roots of polynomials {G}_{n}^{(k)}(x). After that, we generalize the obtained results for all positive real numbers *k*. When k=2, we write

where

and *a* is a positive real number. Now, we can give the following lemma to be used the later.

**Lemma 3.1** *If* *r* *is a maximal root of a function* *f* *with positive leading coefficient*, *then* f(x)>0 *for all* x>r. *Conversely*, *if* f(x)>0 *for all* x\ge t, *then* r<t. *If* f(s)<0 , *then* s<r [2].

**Lemma 3.2** *For* n\ge 2, {G}_{n}^{(2)}(x) *has at least one real root on the interval* (a,a+1) *and* {g}_{n}\in (a,a+1), *where* {g}_{n} *is the maximal root of polynomial* {G}_{n}^{(2)}(x).

*Proof* Some of polynomials {G}_{n}^{(2)}(x) are as follows:

Note that polynomials {G}_{n}^{(2)}(x) are monic polynomials with degree *n* and constant term −*a*. If we write for x=a, then we have

For k\ge 2, if we suppose {G}_{k}^{(2)}(a)\le {a}^{2}{G}_{k-1}^{(2)}(a)<0, then by using the recursive relation (14), we get

Thus, for x=a, we get {G}_{n}^{(2)}(x)<0. Similarly, when x=a+1, we have {G}_{n}^{(2)}(x)>0. Therefore, {G}_{n}^{(2)}(x) has at least one real root on the interval (a,a+1), and we write {g}_{n}\in (a,a+1) for the maximal root of {G}_{n}^{(2)}(x), which results easily from Lemma 3.1 and the recursive relation for {G}_{n}^{(2)}(x). □

Let {g}_{n} denote the maximal root of polynomial {G}_{n}^{(2)}(x) for every n\in \mathbb{N}. Then we can give the following proposition to illustrate the monotonicity of \{{g}_{2n-1}\} and \{{g}_{2n}\}.

**Proposition 3.3** *The sequence* \{{g}_{2n-1}\} *is a monotonically increasing sequence and the sequence* \{{g}_{2n}\} *is a monotonically decreasing sequence*.

*Proof* Firstly, we consider polynomials {G}_{n}^{(2)}(x) with odd indices. By a direct computation, we get {G}_{3}^{(2)}(a)=-{a}^{3}<0, {g}_{3}>a, a={g}_{1}. Assume that {g}_{1}<{g}_{3}<{g}_{5}<\cdots <{g}_{2k-3}<{g}_{2k-1}. We can write {G}_{2k-3}^{(2)}({g}_{2k-1})>0. Thus, it can be easily seen that

By using equation (15), we can write

So, from equation (16) we write

Therefore, polynomials {G}_{2k+1}^{(2)}(x) must have a root greater than {g}_{2k-1}. So, we get

After that we consider polynomials {G}_{n}^{(2)}(x) with even indices. From the recursive relation (14), we can obtain

Since {G}_{2k-1}^{(2)}({g}_{2k-1})=0, by using Lemma 3.1, we can get {G}_{2k+1}^{(2)}({g}_{2k-1})<0. Thus, we get

Again, by using Lemma 3.1, we can write

From the recursive relation (14), we can write

From equation (22), we can get

So, we have {g}_{2k}<{g}_{2k-2}. Thus, \{{g}_{2n-1}\} is a monotonically increasing sequence and bounded above by the number a+1. Similarly, \{{g}_{2n}\} is a monotonically decreasing sequence and bounded below by the number *a*. If we denote the {lim}_{x\to \mathrm{\infty}}{g}_{2n-1} by {g}_{\mathrm{odd}} and {lim}_{x\to \mathrm{\infty}}{g}_{2n} by {g}_{\mathrm{even}}, then we can write {g}_{\mathrm{odd}}={g}_{\mathrm{even}}. □

**Proposition 3.4** *For polynomials* {G}_{2n-1}^{(2)}(x) *and* {G}_{2n}^{(2)}(x), *the sequences* \{{g}_{2n-1}\} *and* \{{g}_{2n}\} *converge to the following number* *ζ*:

*Proof* Using the Binet formula of relation (14), for all [a,a+1], we can see that \alpha (x)\ge \alpha (a)>1 and |\beta (x)|=\frac{1}{\alpha (x)}\le \frac{1}{\alpha (a)}. Thus, we get

If we write n=2k-1 and x={g}_{2k-1} in equation (5), then we have

And from equation (25) we write

A(x) and B(x) are continuous on the interval [a,a+1], this implies that |A(x)| and |B(x)| are bounded below and above on [a,a+1]. So, since a\ge 1, we get

From Binet formula (5), we have

Also, by the aid of similar discussion, if we take n=2k and x={g}_{2k}, then we find that

That is,

Notice that if we take a=1 in equation (29), then our result coincides with the result of Tewodros [7]. □

For *ζ* numbers in equation (23), from Proposition 3.4 we can deduce the following result.

**Corollary 3.5** *For every positive integer* *a*, *we have*

Now, we give a proposition for the maximal real roots of {G}_{n}^{(k)}(x) without proof.

**Proposition 3.6** *The maximal real roots of* {G}_{n}^{(k)}(x) *provide the following equation*:

*where the numbers* g={g}_{n}={g}_{n}(k) *are the maximal real roots of* {G}_{n}^{(k)}(x), *that is*,

*which implies*

*whenever* k>2.

*and*

*whenever* a>1 *for every* n\in \mathbb{N}.

*Proof* The proof can be easily seen as being similar to the proof of Proposition 3.4 □

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## Acknowledgements

The authors are very grateful to the referees for very helpful suggestions and comments about the paper which improved the presentation and its readability.

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Halıcı, S., Akyüz, Z. A note on the roots of some special type polynomials.
*J Inequal Appl* **2013**, 466 (2013). https://doi.org/10.1186/1029-242X-2013-466

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DOI: https://doi.org/10.1186/1029-242X-2013-466