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A note on the roots of some special type polynomials
Journal of Inequalities and Applicationsvolume 2013, Article number: 466 (2013)
In this study, we investigate the polynomials for and positive integers k and a positive real number a, with the initial values ,
We give some fundamental properties related to them. Also, we obtain asymptotic results for the roots of polynomials .
The polynomials defined by Catalan, for , as follows
are called Fibonacci polynomials and denoted by , . The Fibonacci-type polynomials , , are defined by
where and are seed polynomials. There are several studies about the properties of zeros of polynomials . 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 with the initial values , . In , the authors generalized Moore’s result for these polynomials. They considered the initial values , , where a and b are integer numbers. In , the author determined the absolute values of complex zeros of these polynomials. In , Ricci studied this problem in the case and . In , 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 and . 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  studied a special case of the polynomials we investigate.
2 Some fundamental properties of polynomials
In this section, we give some fundamental properties of polynomials , for , defined by the recursive formula as follows:
The characteristic equation for (4) is and its roots are
Note that , and . For relation (4), the Binet formula is
Proposition 2.1 For , the generating function for polynomials is
Proof Let be the generating function for polynomials . So, we write
If we multiply both sides of equation (8) by and , respectively, then we can get
The last two equations give us the following equation:
If we use the recurrence relation and simplify it, we write
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 , we have
In the following propositions, we give some sums formulas related to polynomials .
Proposition 2.3 For , we have
Proof Proof of formula (10) follows now immediately from (7). □
Proposition 2.4 For , we have the following sum formulas:
Proof From the Binet formula, we can write
where , . If we substitute the equations
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 , , 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
In this section, firstly for , we investigate the roots of polynomials . After that, we generalize the obtained results for all positive real numbers k. When , we write
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 for all . Conversely, if for all , then . If , then .
Lemma 3.2 For , has at least one real root on the interval and , where is the maximal root of polynomial .
Proof Some of polynomials are as follows:
Note that polynomials are monic polynomials with degree n and constant term −a. If we write for , then we have
For , if we suppose , then by using the recursive relation (14), we get
Thus, for , we get . Similarly, when , we have . Therefore, has at least one real root on the interval , and we write for the maximal root of , which results easily from Lemma 3.1 and the recursive relation for . □
Let denote the maximal root of polynomial for every . Then we can give the following proposition to illustrate the monotonicity of and .
Proposition 3.3 The sequence is a monotonically increasing sequence and the sequence is a monotonically decreasing sequence.
Proof Firstly, we consider polynomials with odd indices. By a direct computation, we get , , . Assume that . We can write . Thus, it can be easily seen that
By using equation (15), we can write
So, from equation (16) we write
Therefore, polynomials must have a root greater than . So, we get
After that we consider polynomials with even indices. From the recursive relation (14), we can obtain
Since , by using Lemma 3.1, we can get . 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 . Thus, is a monotonically increasing sequence and bounded above by the number . Similarly, is a monotonically decreasing sequence and bounded below by the number a. If we denote the by and by , then we can write . □
Proposition 3.4 For polynomials and , the sequences and converge to the following number ζ:
Proof Using the Binet formula of relation (14), for all , we can see that and . Thus, we get
If we write and in equation (5), then we have
And from equation (25) we write
and are continuous on the interval , this implies that and are bounded below and above on . So, since , we get
From Binet formula (5), we have
Also, by the aid of similar discussion, if we take and , then we find that
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 without proof.
Proposition 3.6 The maximal real roots of provide the following equation:
where the numbers are the maximal real roots of , that is,
whenever for every .
Proof The proof can be easily seen as being similar to the proof of Proposition 3.4 □
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The authors are very grateful to the referees for very helpful suggestions and comments about the paper which improved the presentation and its readability.
The authors declare that they have no competing interests.
All authors completed the paper together. All authors read and approved the final manuscript.