A note on the roots of some special type polynomials
© Halıcı and Akyüz; licensee Springer. 2013
Received: 18 October 2012
Accepted: 20 September 2013
Published: 7 November 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 .
KeywordsFibonacci polynomials Binet formula generating function
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
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)
In the following propositions, we give some sums formulas related to polynomials .
Proof Proof of formula (10) follows now immediately from (7). □
Thus, the proof is completed. Similarly, the second part of the proposition can be seen. □
3 Asymptotic behaviors of the maximal roots for polynomials
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 .
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.
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 . □
For ζ numbers in equation (23), from Proposition 3.4 we can deduce the following result.
Now, we give a proposition for the maximal real roots of without proof.
whenever for every .
Proof The proof can be easily seen as being similar to the proof of Proposition 3.4 □
The authors are very grateful to the referees for very helpful suggestions and comments about the paper which improved the presentation and its readability.
- Koshy T: Fibonacci and Lucas Numbers with Applications. Wiley, New York; 2001.MATHView ArticleGoogle Scholar
- Matyas F: Bounds for the zeros of Fibonacci-like polynomials. Acta Acad. Paedagog. Agriensis Sect. Mat. 1998, 25: 15–20.MATHMathSciNetGoogle Scholar
- Moore GA: The limit of the golden numbers is . Fibonacci Q. 1993, 32(3):211–217.Google Scholar
- Prodinger H: The asymptotic behavior of the golden numbers. Fibonacci Q. 1996, 34(3):224–225.MATHMathSciNetGoogle Scholar
- Zhou J-R, Srivastava HM, Wang ZG: Asymptotic distributions of the zeros of a family of hypergeometric polynomials. Proc. Am. Math. Soc. 2012, 140: 2333–2346. 10.1090/S0002-9939-2011-11117-1MATHMathSciNetView ArticleGoogle Scholar
- Ricci PE: Generalized Lucas polynomials and Fibonacci polynomials. Riv. Mat. Univ. Parma 1995, 4: 137–146.MATHMathSciNetGoogle Scholar
- Amdeberhan T: A note on Fibonacci-type polynomials. Integers 2010, 10: 13–18.MATHMathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.