A note on the roots of some special type polynomials

R E S E A R C H

Open Access

A note on the roots of some special type

polynomials

Serpil Halıcı

_{and Zeynep Akyüz}

*_{Correspondence:}

shalici@sakarya.edu.tr Department of Mathematics, Faculty of Arts and Science, Sakarya University, Sakarya, 54187, Turkey

Abstract

In this study, we investigate the polynomials forn≥2 and positive integerskand a positive real numbera, with the initial valuesG0(x) = –a,G1(x) =x–a

G_n(k₊₂) (x) =xkG(_nk₊₁) (x) +G(_nk)(x).

We give some fundamental properties related to them. Also, we obtain asymptotic results for the roots of polynomialsG(_nk)(x).

MSC: 11B39; 11B37

Keywords: Fibonacci polynomials; Binet formula; generating function

1 Introduction

The polynomials defined by Catalan, forn≥, as follows

Fn+(x) =xFn+(x) +Fn(x); F(x) = , F(x) =x ()

are called Fibonacci polynomials and denoted byFn(x), []. The Fibonacci-type polyno-mialsGn(x),n≥, are defined by

Gn+(x) =xGn+(x) +Gn(x), ()

whereG(x) andG(x) are seed polynomials. There are several studies about the prop-erties of zeros of polynomialsGn(x). However, there are no general formulas for zeros of Fibonacci-type polynomials. In [, ], the authors studied the limiting behavior of the maximal real roots of polynomialsGn(x) with the initial valuesG(x) = –,G(x) =x– . In [], the authors generalized Moore’s result for these polynomials. They considered the initial valuesG(x) =a,G(x) =x+b, whereaandbare integer numbers. In [], the au-thor determined the absolute values of complex zeros of these polynomials. In [], Ricci studied this problem in the casea=  andb= . In [], Tewodros investigated the conver-gence of maximal real roots of different Fibonacci-type polynomials given by the following relation:

G(k)_n+(x) =xkG(k)_n+(x) +G(k)_n (x), n≥, ()

wherek is a positive integer number. The initial values of the recursive relation () are

G(k)

 (x) = – andG (k)

 (x) =x– . In this study, firstly, we investigate some fundamental properties of Fibonacci-type polynomials. We give some combinatorial identities related to equation (). Then, we investigate the limit of maximal real roots of these polynomials. We notice that Tewodros [] studied a special casea=  of the polynomials we investigate. 2 Some fundamental properties of polynomialsG(k)

n (x)

In this section, we give some fundamental properties of polynomialsG(k)_n (x), forn≥, defined by the recursive formula as follows:

G(k)_n+(x) =xkG(k)_n+(x) +G(k)_n (x); G(k)_ (x) = –a; G(k)_ (x) =x–a. () The characteristic equation for () ist–xkt–  =  and its roots are

α(x) =x

k₊√_xk_{+ } 

and

β(x) = x

k_–√_xk_{+ }

 .

Note thatα(x)β(x) = –,α(x) +β(x) =xk_{andα(x) –β(x) =}√_xk_{+ . For relation (), the} Binet formula is

G(k)_n (x) =A(x)αn(x) +B(x)βn(x), () where

A(x) =(x–a) +ax

k_–a√_xk_{+ }

√xk_{+ } , B(x) =

–(x–a) –axk–a√xk_{+ }

√xk_{+ } . ()

Proposition . For n≥,the generating function for polynomials G(k)n (x)is

H_r(k)(x,t) = n≥

G(k)_n+r(x)tn=

⎧ ⎨ ⎩

G(k)r (x)+G(k)r–(x)t

–xk_t–t , r= , , , . . . , t(axk+x–a)

–xk_t–t , r= .

()

Proof LetHr(k)(x,t) be the generating function for polynomialsG(k)n+r(x). So, we write

H_r(k)(x,t) = n≥

G(k)_n+r(x)tn. ()

If we multiply both sides of equation () byxk_tandt_{, respectively, then we can get}

xktH_r(k)(x,t) =xkG_r(k)(x)t+xkG_r+(k)(x)t+xkG(k)_r+(x)t+· · ·+xkG(k)_r+n–(x)tn+· · · and

The last two equations give us the following equation:

H_r(k)(x,t) –xktH_r(k)(x,t) –tH_r(k)(x,t) =G(k)_r (x)t+G(k)_r+(x) –xkG(k)_r (x)t

+G(k)_r+(x) –xkG(k)_r+(x) –G(k)_r (x)t+· · · +G(k)_n+r(x) –xkG(k)_n+r–(x) –G(k)_n+r–(x)tn+· · ·. If we use the recurrence relation and simplify it, we write

H_r(k)(x,t) –xktH_r(k)(x,t) –tH_r(k)(x,t) =G(k)_r (x)t+G(k)r+(x) –xkG(k)r (x)

t,

i.e.,

H_rk(x,t) =

⎧ ⎨ ⎩

G(k)r (x)+G(k)r–(x)t

–xk_t–t , r= , , , . . . , t(axk+x–a)

–xk_t–t , r= .

Thus, the proof is completed.

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

Proposition .(Cassini-like) For n≥,we have

G(k)_n–(x)G(k)_n+(x) –G(k)_n (x) = (–)n–A(x)B(x) , ()

where

A(x) =(x–a) +ax

k_–a√_xk_{+ } √xk_{+ }

and

B(x) =–(x–a) –ax

k_–a√_xk_{+ }

√xk_{+ } .

In the following propositions, we give some sums formulas related to polynomials

G(k)n (x).

Proposition . For N≥,we have

H_(k)(x, ) –H_N+(k) (x, ) = N

r=

G(k)_r (x) =a–x–ax k_+G(k)

N+(x) +G (k) N (x)

xk . ()

Proof Proof of formula () follows now immediately from ().

Proposition . For N≥,we have the following sum formulas: N

r=

G(k)_r(x) = –x

k+_–axk_(xk_{– ) –G}(k) N(x) +G

(k) N+(x)

and

N

r=

G(k)_r+(x) =ax k_–G(k)

N+(x) +G (k) N+(x)

xk . ()

Proof From the Binet formula, we can write N

r=

G(k)_r(x) =

x–a+aβ

α–β

N

r= αr–

x–a+aα

α–β

N

r=

βr, ()

whereα=α(x),β=β(x). If we substitute the equations N

r=

αr= –α N+

 –α , N

r=

βr= –β N+

 –β

and

 –α –β= –xk

into equation (), then we can get

N

r=

G(k)_r(x) =

x–a+aβ

α–β

 –αN+  –α –

x–a+aα

α–β

 –βN+  –β .

If we rearrange the last equation, then we have

N

r=

G(k)_r(x) = –a(α–β) + (x–a)(α

_–β_{) +a(α}_–β₎ ( –α_{)( –β}_)(α–β)

–[(x–a+aβ)α

N+_{– (x–a+aα)β}N+_] ( –α_{)( –β}_)(α–β)

+α

_β_{[(x–a+aβ)α}N_{– (x–a+aα)β}N_] ( –α_{)( –β}_)(α–β) .

By taking aid of the Binet formula, we can write

N

r=

G(k)_r(x) = –a+ (x–a)(α+β) +a(α

_+αβ+β₎

( –α_{)( –β}₎

+ G

(k) N(x) ( –α_{)( –β}₎–

G(k)_N+(x) ( –α_{)( –β}₎.

If we substituteα+β=xk_{,αβ= –, ( –α}_{)( –β}_{) = –x}k_{into the last equation, we obtain} the following equation:

N

r=

G(k)_r(x) = x

k+_–axk_+axk_–G(k)

N+(x) +G (k) N(x)

–xk .

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

3 Asymptotic behaviors of the maximal roots for polynomialsG(k) n (x)

In this section, firstly fork= , we investigate the roots of polynomialsG(k)n (x). After that, we generalize the obtained results for all positive real numbersk. Whenk= , we write

G()_n+(x) =xG_n+()(x) +G()_n (x), () where

G()_ (x) = –a, G()_ (x) =x–a

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

Lemma . If r is a maximal root of a function f with positive leading coefficient,then f(x) > for all x>r.Conversely,if f(x) > for all x≥t,then r<t.If f(s) <  ,then s<r[].

Lemma . For n≥,G()n (x)has at least one real root on the interval(a,a+ )and gn∈ (a,a+ ),where gnis the maximal root of polynomial G()n (x).

Proof Some of polynomialsG()n (x) are as follows:

G()_ (x) =x–ax–a,

G()_ (x) =x–ax–ax+x–a,

G()_ (x) =x–ax–ax+ x– ax–a, ..

.

Note that polynomialsG()n (x) are monic polynomials with degreenand constant term –a. If we write forx=a, then we have

G()_ (a) = ,

G()_ (a) = –a< ,

G()_ (a) = –a=aG()_ (a) < ,

G()_ (a) = –a–a≤–a=aG()_ (a) < , ..

.

Fork≥, if we supposeG()_k (a)≤aG()_k–(a) < , then by using the recursive relation (), we get

G()_k+(a) =aG_k()(a) +G()_k–(a) < .

Thus, forx=a, we getG()n (x) < . Similarly, whenx=a+ , we haveG()n (x) > . Therefore,

G()n (x) has at least one real root on the interval (a,a+ ), and we writegn∈(a,a+ ) for the maximal root ofG()n (x), which results easily from Lemma . and the recursive relation

Letgndenote the maximal root of polynomialGn()(x) for everyn∈N. Then we can give the following proposition to illustrate the monotonicity of{gn–}and{gn}.

Proposition . The sequence{gn–}is a monotonically increasing sequence and the

se-quence{gn}is a monotonically decreasing sequence.

Proof Firstly, we consider polynomialsG()n (x) with odd indices. By a direct computation, we getG()_ (a) = –a_{< ,g}

>a,a=g. Assume thatg<g<g<· · ·<gk–<gk–. We can writeG()_k–(gk–) > . Thus, it can be easily seen that

G()_n+k(gn) = (–)k+G()_n–k(gn). ()

By using equation (), we can write

G()_k+(gk–) =G_{(k–)+}() (gk–) = –G()_{(k–)–}(gk–) = –G()_k–(gk–). () So, from equation () we write

G()_k+(gk–) < . ()

Therefore, polynomialsG()_k+(x) must have a root greater thangk–. So, we get

gk+>gk–. ()

After that we consider polynomials G()n (x) with even indices. From the recursive rela-tion (), we can obtain

G()_k+(gk–) =gk– G ()

k(gk–) +G()k–(gk–). ()

SinceG()_k–(gk–) = , by using Lemma ., we can getG()_k+(gk–) < . Thus, we get

gk–<gk. () Again, by using Lemma ., we can write

G()_k–(gk) > . ()

From the recursive relation (), we can write

G()_k(gk) =g_k G_k–() (gk) +G()_k–(gk). () From equation (), we can get

–g_kG()_k–(gk) =G()_k–(gk) < .

bounded below by the numbera. If we denote thelim_x→∞gn–bygoddandlimx→∞gnby

geven, then we can writegodd=geven.

Proposition . For polynomials G_n–() (x)and G()_n(x),the sequences{gn–}and{gn}

con-verge to the following numberζ:

ζ =

( –a₎_{+ a}_{– ( –a}₎

a . ()

Proof Using the Binet formula of relation (), for all [a,a+ ], we can see thatα(x)≥ α(a) >  and|β(x)|=_α(x) ≤_α(a) . Thus, we get

lim

n→∞α

n_{(x) = +∞; lim}

n→∞β

n_{(x) = . ()}

If we writen= k–  andx=gk–in equation (), then we have

A(gk–)αk–(gk–) +B(gk–)βk–(gk–) = . ()

And from equation () we write

A(gk–) = –B(gk–)

βk–_{(gk–)} αk–_{(gk–)}

. ()

A(x) andB(x) are continuous on the interval [a,a+ ], this implies that|A(x)|and|B(x)| are bounded below and above on [a,a+ ]. So, sincea≥, we get

lim

k→∞A(gk–) =A(godd) = . ()

From Binet formula (), we have

lim k→∞gk–=

( –a₎_{+ a}_{– ( –a}₎

a . ()

Also, by the aid of similar discussion, if we taken= kandx=gk, then we find that

lim k→∞gk=

( –a₎

+ a_{– ( –a}₎

a .

That is,

ζ =

( –a₎

+ a_{– ( –a}₎

a . ()

Notice that if we takea=  in equation (), then our result coincides with the result of

Tewodros [].

Corollary . For every positive integer a,we have

a<ζ<a+ . () Now, we give a proposition for the maximal real roots ofG(k)n (x) without proof.

Proposition . The maximal real roots of G(k)n (x)provide the following equation:

g– a+agk–agk–= , ()

where the numbers g=gn=gn(k)are the maximal real roots of G(k)n (x),that is,

agn=a–gn–k+ agn–k, ()

which implies a

 +a(a+ )k–<gn(k) –a<

a

 +ak,

whenever k> .

a– 

a(a+ ) <

a–gn()

agn() =gn() –a< 

a+ 

and

lim

k→∞gn(k) =a,

whenever a> for every n∈N.

Proof The proof can be easily seen as being similar to the proof of Proposition .

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors completed the paper together. All authors read and approved the final manuscript.

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.

Received: 18 October 2012 Accepted: 20 September 2013 Published:07 Nov 2013

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