Open Access

A note on the roots of some special type polynomials

Journal of Inequalities and Applications20132013:466

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

Received: 18 October 2012

Accepted: 20 September 2013

Published: 7 November 2013

Abstract

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

G n + 2 ( k ) ( x ) = x k G n + 1 ( k ) ( x ) + G n ( k ) ( x ) .

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.

Keywords

Fibonacci polynomialsBinet formulagenerating function

1 Introduction

The polynomials defined by Catalan, for n 0 , as follows
F n + 2 ( x ) = x F n + 1 ( x ) + F n ( x ) ; F 1 ( x ) = 1 , F 2 ( x ) = x
(1)
are called Fibonacci polynomials and denoted by F n ( x ) , [1]. The Fibonacci-type polynomials G n ( x ) , n 0 , are defined by
G n + 2 ( x ) = x G n + 1 ( x ) + G n ( x ) ,
(2)
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:
G n + 2 ( k ) ( x ) = x k G n + 1 ( k ) ( x ) + G n ( k ) ( x ) , n 0 ,
(3)

where k is a positive integer number. The initial values of the recursive relation (3) are G ( k ) 0 ( x ) = 1 and G ( k ) 1 ( 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 ( k ) n ( x ) , for n 0 , defined by the recursive formula as follows:
G n + 2 ( k ) ( x ) = x k G n + 1 ( k ) ( x ) + G n ( k ) ( x ) ; G 0 ( k ) ( x ) = a ; G 1 ( k ) ( x ) = x a .
(4)
The characteristic equation for (4) is t 2 x k t 1 = 0 and its roots are
α ( x ) = x k + x 2 k + 4 2
and
β ( x ) = x k x 2 k + 4 2 .
Note that α ( x ) β ( x ) = 1 , α ( x ) + β ( x ) = x k and α ( x ) β ( x ) = x 2 k + 4 . For relation (4), the Binet formula is
G n ( k ) ( x ) = A ( x ) α n ( x ) + B ( x ) β n ( x ) ,
(5)
where
A ( x ) = 2 ( x a ) + a x k a x 2 k + 4 2 x 2 k + 4 , B ( x ) = 2 ( x a ) a x k a x 2 k + 4 2 x 2 k + 4 .
(6)
Proposition 2.1 For n 0 , the generating function for polynomials G n ( k ) ( x ) is
H r ( k ) ( x , t ) = n 0 G n + r ( k ) ( x ) t n = { G r ( k ) ( x ) + G r 1 ( k ) ( x ) t 1 x k t t 2 , r = 1 , 2 , 3 , , t ( a x k + x a ) 1 x k t t 2 , r = 0 .
(7)
Proof Let H r ( k ) ( x , t ) be the generating function for polynomials G n + r ( k ) ( x ) . So, we write
H r ( k ) ( x , t ) = n 0 G n + r ( k ) ( x ) t n .
(8)
If we multiply both sides of equation (8) by x k t and t 2 , respectively, then we can get
x k t H r ( k ) ( x , t ) = x k G r ( k ) ( x ) t 1 + x k G r + 1 ( k ) ( x ) t 2 + x k G r + 2 ( k ) ( x ) t 3 + + x k G r + n 1 ( k ) ( x ) t n +
and
t 2 H r ( k ) ( x , t ) = G r ( k ) ( x ) t 2 + G r + 1 ( k ) ( x ) t 3 + G r + 2 ( k ) ( x ) t 4 + + G r + n 2 ( k ) ( x ) t n + .
The last two equations give us the following equation:
H r ( k ) ( x , t ) x k t H r ( k ) ( x , t ) t 2 H r ( k ) ( x , t ) = G r ( k ) ( x ) t 0 + ( G r + 1 ( k ) ( x ) x k G r ( k ) ( x ) ) t + ( G r + 2 ( k ) ( x ) x k G r + 1 ( k ) ( x ) G r ( k ) ( x ) ) t 2 + + ( G n + r ( k ) ( x ) x k G n + r 1 ( k ) ( x ) G n + r 2 ( k ) ( x ) ) t n + .
If we use the recurrence relation and simplify it, we write
H r ( k ) ( x , t ) x k t H r ( k ) ( x , t ) t 2 H r ( k ) ( x , t ) = G r ( k ) ( x ) t 0 + ( G r + 1 ( k ) ( x ) x k G r ( k ) ( x ) ) t ,
i.e.,
H r k ( x , t ) = { G r ( k ) ( x ) + G r 1 ( k ) ( x ) t 1 x k t t 2 , r = 1 , 2 , 3 , , t ( a x k + x a ) 1 x k t t 2 , r = 0 .

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 0 , we have
G n 1 ( k ) ( x ) G n + 1 ( k ) ( x ) [ G n ( k ) ( x ) ] 2 = ( 1 ) n 1 [ A ( x ) B ( x ) ] ,
(9)
where
A ( x ) = 2 ( x a ) + a x k a x 2 k + 4 2 x 2 k + 4
and
B ( x ) = 2 ( x a ) a x k a x 2 k + 4 2 x 2 k + 4 .

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

Proposition 2.3 For N 0 , we have
H 0 ( k ) ( x , 1 ) H N + 1 ( k ) ( x , 1 ) = r = 0 N G r ( k ) ( x ) = 2 a x a x k + G N + 1 ( k ) ( x ) + G N ( k ) ( x ) x k .
(10)

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

Proposition 2.4 For N 0 , we have the following sum formulas:
r = 0 N G 2 r ( k ) ( x ) = x k + 1 a x k ( x k 1 ) G 2 N ( k ) ( x ) + G 2 N + 2 ( k ) ( x ) x 2 k
(11)
and
r = 0 N G 2 r + 1 ( k ) ( x ) = a x k G 2 N + 1 ( k ) ( x ) + G 2 N + 3 ( k ) ( x ) x 2 k .
(12)
Proof From the Binet formula, we can write
r = 0 N G 2 r ( k ) ( x ) = ( x a + a β α β ) r = 0 N α 2 r ( x a + a α α β ) r = 0 N β 2 r ,
(13)
where α = α ( x ) , β = β ( x ) . If we substitute the equations
r = 0 N α 2 r = 1 α 2 N + 2 1 α 2 , r = 0 N β 2 r = 1 β 2 N + 2 1 β 2
and
( 1 α 2 ) ( 1 β 2 ) = x 2 k
into equation (13), then we can get
r = 0 N G 2 r ( k ) ( x ) = ( x a + a β α β ) 1 α 2 N + 2 1 α 2 ( x a + a α α β ) 1 β 2 N + 2 1 β 2 .
If we rearrange the last equation, then we have
r = 0 N G 2 r ( k ) ( x ) = a ( α β ) + ( x a ) ( α 2 β 2 ) + a ( α 3 β 3 ) ( 1 α 2 ) ( 1 β 2 ) ( α β ) [ ( x a + a β ) α 2 N + 2 ( x a + a α ) β 2 N + 2 ] ( 1 α 2 ) ( 1 β 2 ) ( α β ) + α 2 β 2 [ ( x a + a β ) α 2 N ( x a + a α ) β 2 N ] ( 1 α 2 ) ( 1 β 2 ) ( α β ) .
By taking aid of the Binet formula, we can write
r = 0 N G 2 r ( k ) ( x ) = a + ( x a ) ( α + β ) + a ( α 2 + α β + β 2 ) ( 1 α 2 ) ( 1 β 2 ) + G 2 N ( k ) ( x ) ( 1 α 2 ) ( 1 β 2 ) G 2 N + 2 ( k ) ( x ) ( 1 α 2 ) ( 1 β 2 ) .
If we substitute α + β = x k , α β = 1 , ( 1 α 2 ) ( 1 β 2 ) = x 2 k into the last equation, we obtain the following equation:
r = 0 N G 2 r ( k ) ( x ) = x k + 1 a x k + a x 2 k G 2 N + 2 ( k ) ( x ) + G 2 N ( k ) ( x ) x 2 k .

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
G n + 2 ( 2 ) ( x ) = x 2 G n + 1 ( 2 ) ( x ) + G n ( 2 ) ( x ) ,
(14)
where
G 0 ( 2 ) ( x ) = a , G 1 ( 2 ) ( x ) = x a

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 t , then r < t . If f ( s ) < 0 , then s < r [2].

Lemma 3.2 For n 2 , G n ( 2 ) ( x ) has at least one real root on the interval ( a , a + 1 ) and g n ( 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:
G 2 ( 2 ) ( x ) = x 3 a x 2 a , G 3 ( 2 ) ( x ) = x 5 a x 4 a x 2 + x a , G 4 ( 2 ) ( x ) = x 7 a x 6 a x 4 + 2 x 3 2 a x 2 a ,
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
G 1 ( 2 ) ( a ) = 0 , G 2 ( 2 ) ( a ) = a < 0 , G 3 ( 2 ) ( a ) = a 3 = a 2 G 2 ( 2 ) ( a ) < 0 , G 4 ( 2 ) ( a ) = a 5 a a 5 = a 2 G 3 ( 2 ) ( a ) < 0 ,
For k 2 , if we suppose G k ( 2 ) ( a ) a 2 G k 1 ( 2 ) ( a ) < 0 , then by using the recursive relation (14), we get
G k + 1 ( 2 ) ( a ) = a 2 G k ( 2 ) ( a ) + G k 1 ( 2 ) ( a ) < 0 .

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 ( 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 N . Then we can give the following proposition to illustrate the monotonicity of { g 2 n 1 } and { g 2 n } .

Proposition 3.3 The sequence { g 2 n 1 } is a monotonically increasing sequence and the sequence { g 2 n } 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 < < g 2 k 3 < g 2 k 1 . We can write G 2 k 3 ( 2 ) ( g 2 k 1 ) > 0 . Thus, it can be easily seen that
G n + k ( 2 ) ( g n ) = ( 1 ) k + 1 G n k ( 2 ) ( g n ) .
(15)
By using equation (15), we can write
G 2 k + 1 ( 2 ) ( g 2 k 1 ) = G ( 2 k 1 ) + 2 ( 2 ) ( g 2 k 1 ) = G ( 2 k 1 ) 2 ( 2 ) ( g 2 k 1 ) = G 2 k 3 ( 2 ) ( g 2 k 1 ) .
(16)
So, from equation (16) we write
G 2 k + 1 ( 2 ) ( g 2 k 1 ) < 0 .
(17)
Therefore, polynomials G 2 k + 1 ( 2 ) ( x ) must have a root greater than g 2 k 1 . So, we get
g 2 k + 1 > g 2 k 1 .
(18)
After that we consider polynomials G n ( 2 ) ( x ) with even indices. From the recursive relation (14), we can obtain
G 2 k + 1 ( 2 ) ( g 2 k 1 ) = g 2 k 1 2 G 2 k ( 2 ) ( g 2 k 1 ) + G 2 k 1 ( 2 ) ( g 2 k 1 ) .
(19)
Since G 2 k 1 ( 2 ) ( g 2 k 1 ) = 0 , by using Lemma 3.1, we can get G 2 k + 1 ( 2 ) ( g 2 k 1 ) < 0 . Thus, we get
g 2 k 1 < g 2 k .
(20)
Again, by using Lemma 3.1, we can write
G 2 k 1 ( 2 ) ( g 2 k ) > 0 .
(21)
From the recursive relation (14), we can write
G 2 k ( 2 ) ( g 2 k ) = g 2 k 2 G 2 k 1 ( 2 ) ( g 2 k ) + G 2 k 2 ( 2 ) ( g 2 k ) .
(22)
From equation (22), we can get
g 2 k 2 G 2 k 1 ( 2 ) ( g 2 k ) = G 2 k 2 ( 2 ) ( g 2 k ) < 0 .

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

Proposition 3.4 For polynomials G 2 n 1 ( 2 ) ( x ) and G 2 n ( 2 ) ( x ) , the sequences { g 2 n 1 } and { g 2 n } converge to the following number ζ:
ζ = ( 1 a 2 ) 2 + 8 a 2 ( 1 a 2 ) 2 a .
(23)
Proof Using the Binet formula of relation (14), for all [ a , a + 1 ] , we can see that α ( x ) α ( a ) > 1 and | β ( x ) | = 1 α ( x ) 1 α ( a ) . Thus, we get
lim n α n ( x ) = + ; lim n β n ( x ) = 0 .
(24)
If we write n = 2 k 1 and x = g 2 k 1 in equation (5), then we have
A ( g 2 k 1 ) α 2 k 1 ( g 2 k 1 ) + B ( g 2 k 1 ) β 2 k 1 ( g 2 k 1 ) = 0 .
(25)
And from equation (25) we write
A ( g 2 k 1 ) = B ( g 2 k 1 ) ( β 2 k 1 ( g 2 k 1 ) α 2 k 1 ( g 2 k 1 ) ) .
(26)
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 1 , we get
lim k A ( g 2 k 1 ) = A ( g odd ) = 0 .
(27)
From Binet formula (5), we have
lim k g 2 k 1 = ( 1 a 2 ) 2 + 8 a 2 ( 1 a 2 ) 2 a .
(28)
Also, by the aid of similar discussion, if we take n = 2 k and x = g 2 k , then we find that
lim k g 2 k = ( 1 a 2 ) 2 + 8 a 2 ( 1 a 2 ) 2 a .
That is,
ζ = ( 1 a 2 ) 2 + 8 a 2 ( 1 a 2 ) 2 a .
(29)

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
a < ζ < a + 1 .
(30)

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:
g 2 a + a g k a 2 g k 1 = 0 ,
(31)
where the numbers g = g n = g n ( k ) are the maximal real roots of G n ( k ) ( x ) , that is,
a g n = a 2 g n 2 k + 2 a g n 1 k ,
(32)
which implies
a 1 + a ( a + 1 ) k 1 < g n ( k ) a < a 1 + a k ,
whenever k > 2 .
a 1 a ( a + 1 ) < 2 a g n ( 2 ) a g n ( 2 ) = g n ( 2 ) a < 1 a + 1
and
lim k g n ( k ) = a ,

whenever a > 1 for every n N .

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

Declarations

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.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Arts and Science, Sakarya University

References

  1. Koshy T: Fibonacci and Lucas Numbers with Applications. Wiley, New York; 2001.MATHView ArticleGoogle Scholar
  2. Matyas F: Bounds for the zeros of Fibonacci-like polynomials. Acta Acad. Paedagog. Agriensis Sect. Mat. 1998, 25: 15–20.MATHMathSciNetGoogle Scholar
  3. Moore GA: The limit of the golden numbers is 3 / 2 . Fibonacci Q. 1993, 32(3):211–217.Google Scholar
  4. Prodinger H: The asymptotic behavior of the golden numbers. Fibonacci Q. 1996, 34(3):224–225.MATHMathSciNetGoogle Scholar
  5. 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
  6. Ricci PE: Generalized Lucas polynomials and Fibonacci polynomials. Riv. Mat. Univ. Parma 1995, 4: 137–146.MATHMathSciNetGoogle Scholar
  7. Amdeberhan T: A note on Fibonacci-type polynomials. Integers 2010, 10: 13–18.MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Halıcı and Akyüz; licensee Springer. 2013

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.