On the 2k step Jordan Fibonacci sequence

R E S E A R C H

Open Access

On the



k

-step Jordan-Fibonacci

sequence

Omur Deveci

_{, Sait Ta¸s}

_{and A Kılıçman}

*_{Correspondence:}

[email protected]

3_{Department of Mathematics,}

University Putra Malaysia, Serdang, Selangor 43400 UPM, Malaysia Full list of author information is available at the end of the article

Abstract

In this paper, we define the 2k-step Jordan-Fibonacci sequence, and then we study the 2k-step Jordan-Fibonacci sequence modulom. Also, we obtain the cyclic groups from the multiplicative orders of the generating matrix of the 2k-step

Jordan-Fibonacci sequence when read modulom, and we give the relationships among the orders of the cyclic groups obtained and the periods of the 2k-step Jordan-Fibonacci sequence modulom. Furthermore, we extend the 2k-step Jordan-Fibonacci sequence to groups, and then we examine this sequence in the finite groups. Finally, we obtain the period of the 2k-step Jordan-Fibonacci sequence in the generalized quaternion groupQ2nas applications of the results produced.

MSC: Primary 11B50; 20F05; secondary 11C20; 20D60

Keywords: Jordan-Fibonacci sequence; group; matrix; period

1 Introduction

Suppose that the (n+k)th term of a sequence is defined consecutively by a linear combi-nation of the precedingkterms:

an+k=can+can++· · ·+ck–an+k–,

wherec,c, . . . ,ck–are real constants. In [], Kalman derived a number of closed-form

formulas for the generalized sequence by the companion matrix method as follows. Let the matrixAbe defined by

A= [aij]k×k=

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

   . . .  

   . . .  

   . ..  

..

. ... ... ... ...

   . . .  

c c c ck– ck–

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

,

then

Thek-step Fibonacci sequence{Fk

n}is defined by initial valuesFk=· · ·=Fkk–= ,Fkk= 

It is clear that the characteristic polynomial of thek-step Fibonacci sequence is as fol-lows:

Pk(x) =xk–xk––· · ·–x– .

A square matrix of the form

⎡

is called a Jordan block. The Jordan block of orderkis denoted byJk(λ). A Jordan matrix

is a block diagonal matrix whose blocks are all Jordan blocks.

Work on Fibonacci sequences in groups began with the previous study of Wall [] where the ordinary Fibonacci sequences in cyclic groups were investigated. Later, Wilcox [] ex-tended the problem to Abelian groups. Recently, many authors have studied some special linear recurrence sequences in algebraic structures; see, for example, [–]. In [, , –]. The authors obtained the cyclic groups via some special matrices.

In this paper, we obtain the (k)×(k) Jordan-type matrixJwith the aid of the character-istic polynomial of thek-step Fibonacci sequence, and then we define the k-step Jordan-Fibonacci sequence by using the Jordan-Jordan-Fibonacci matrixFk

J of order kwhich is produced

from the matrixJ. Then we study the k-step Jordan-Fibonacci sequence modulom, and we obtain the cyclic groups from the multiplicative orders of the Jordan-Fibonacci matrix

obtain the period of the k-step Jordan-Fibonacci sequence in the generalized quaternion groupQnas applications of the results produced.

2 Main results and proofs

We consider the (k)×(k) Jordan-type matrixJwhich is defined by using the characteristic polynomial of thek-step Fibonacci sequence:

⎡

Next, we define the k-step Jordan-Fibonacci sequence with the aid of the matrixFk J as

follows:

ak_n+k=ak_n+k–+ak_n+k–ak_n+k––· · ·–ak_n ()

forn≥, with initial conditionsak_=· · ·=ak_k–=  andak_k= . By mathematical induction onα, we may write

and

Now we consider the k-step Jordan-Fibonacci sequence{ak

n}modulom. If we reduce

the sequence{ak

n}by a modulusm, then we get the repeating sequence, denoted by

ak_n(m)=ak_(m),ak_(m), . . . ,ak_i(m), . . .,

where we denoteak

i (modm) byaki(m). It has the same recurrence relation as in ().

A sequence is said to beperiodicif, after a certain point, it consists only of repetitions of a fixed subsequence. The number of elements in the shortest repeating subsequence is called the period of the sequence. In particular, if the firstnelements in the sequence form a repeating subsequence, then this sequence issimply periodicand its period isn.

Theorem . {ak

n(m)}forms a simply periodic sequence.

Proof LetX={(x,x, . . . ,xk)|xi’s be integers such that ≤xi≤m– }. Since there are

mk_{distinct k-tuples of elements ofZ}

m, at least one of the k-tuples appears twice in the

sequence{ak

n(m)}. Therefore, the subsequence following this k-tuple repeats; hence, the

sequence is periodic. Assume thatu>vand

ak_u+(m) =ak_v+(m),ak_u+(m) =ak_v+(m), . . . ,ak_u+k(m) =ak_v+k(m),

thenu≡v(modk). By (), we may write

ak_n= –ak_n+k+ak_n+k–+ak_n+k–ak_n+k––· · ·–ak_n+.

Then we obtain

ak_u(m) =ak_v(m),ak_u–(m) =ak_v–(m), . . . ,ak_u–v+(m) =ak_(m).

Thus it is verified that the sequence is simply periodic.

The period of the sequence{ak

Given an integer matrixA= [aij],A(modm) means that all entries ofAare modulom,

that is,

A(modm) = aij(modm)

.

Let us consider the set

Am=

Ai(modm)|i≥.

Ifgcd(m,detA) = , thenAm is a cyclic group; ifgcd(m,detA)= , thenAmis a

semi-group. Let the notation|Am|denote the order of the setAm. SincedetFJk= ,|FJkm|

is a cyclic group for every positive integer m. From () and (), it is easy to see that

Pk_{(m) =|F}k Jm|.

Now we give some properties of the periodPk_{(m) by the following theorems.}

Theorem . Let t be a prime and suppose that u is the largest positive integer with Pk_{(t) =}

Pk_(tu_).Then

Pk _tv_=tv–u_·Pk_(t)

for every v≥u.In particular if Pk_(t_)=Pk_{(t),then P}k_(tv_{) =t}v–_·Pk_(t).

Proof SincePk_{(m) =|F}k

Jm|, we have a positive integernsuch that

F_JkPk(tn+)≡I modtn+,

whereIis the (k)×(k) identity matrix. Then it is clear that

F_JkPk(tn+)≡I modtn,

which implies thatPk_(tn+_{) is divisible byP}k_(tn_{). Furthermore, if we denote}

F_JkPk(tn)=I+ f_ij(t)·tn,

then by the binomial expansion, we may write

F_JkPk(tn)·t= I+ f_ij(t)·tnt

=

t

i=

t i f

(t)

ij ·tn

i_≡

I modtn+.

This yields thatPk(tn+) dividesPk(tn)·t. Then it is clear that

Pk tn+=Pk tn

or

It is easy to see that the latter holds if and only if there isf_ij(t)which is not divisible byt. Sinceuis the largest positive integer such that

Pk(t) =Pk tu,

Pk tu=Pk tu+,

then there isf_ij(u+)which is not divisible byt. Therefore, we obtain

Pk tu+=Pk tu+, and so

Pk tn+=Pk tn+·t=Pk tn·t.

To complete the proof we may use an inductive method.

Theorem . If m has the prime factorization m=u_i=tri

i, where(u≥), then Pk(m)

equals the least common multiple of Pk_(tri i)s.

Proof SincePk_(tri

i) is a period of the sequence{akn(t ri

i)}, it is easy to see that the sequence

{ak n(t

ri

i )}repeats only after blocks of lengthn·Pk(t ri

i), (n∈N). Since alsoPk(m) is a period

of the sequence{ak

n(m)}, the sequence{akn(t ri

i )}repeats after termsPk(m) for all valuesi.

Then we get thatPk_{(m) is the formn·P}k_(pri

i) for all values ofi, and since any such number

gives a period ofPk_{(m), we conclude that}

Pk(m) =lcmPk tr



, . . . ,Pk tru u

.

Now we extend the k-step Jordan-Fibonacci sequence tok-generator groups such that

k≥. LetGbe a finitek-generator group and suppose that

X=(x,x, . . . ,xk)∈G ×G× · · · × G k

|{x,x, . . . ,xk}

=G.

We call (x,x, . . . ,xk) a generatingk-tuple forG.

Definition . For ak-tuple (x,x, . . . ,xk)∈X, we define the Jordan-Fibonacci orbit

JO(G;x,x, . . . ,xk) =

bk_i

by

bk_=x, . . . ,bkk=xk,bkk+= (x)–(x)–· · ·(xk)–,

bk_k+=· · ·=bk_k= (x)–(x)–· · ·(xk)–(xk+)

and

bk_n+k= bk_n–· · · bk_n+k–– bk_n+k bk_n+k–

Theorem . If the group G is finite,then the Jordan-Fibonacci orbitJO(G;x,x, . . . ,xk)

is simply periodic.

Proof Letβbe order of the groupG, then it is clear that there areβkdistinct k-tuples of elements ofG. Thus it is verified that at least one of the k-tuples appears twice in the Jordan-Fibonacci orbitJO(G;x,x, . . . ,xk). Because of the repeating, the sequence is

pe-riodic. Since the Jordan-Fibonacci orbitJO(G;x,x, . . . ,xk) is periodic, there exist natural

numbersmandlwithl≡m(modk) such that

bk_m+=bk_l+,bk_m+=bk_l+, . . . ,bk_m+k=bk_l+k.

From the definition of the Jordan-Fibonacci orbitJO(G;x,x, . . . ,xk), we may write

bk_n= bk_n+–· · · bk_n+k–– bk_n+k bk_n+k– bk_n+k–.

Thus it is verified that

bk_m=bk_l,bk_m–=bk_l–, . . . ,bk_=bk_l–m+.

So the proof is complete.

The period of the Jordan-Fibonacci orbit JO(G;x,x, . . . ,xk) is denoted byPJO(G;x, x, . . . ,xk).

It is well known that the generalized quaternion groupQnis defined by presentation

x,y|xn–=e,y=xn–,y–xy=x–,

forn≥. We will now address the Jordan-Fibonacci orbit of the generalized quaternion groupQnfor generating pair (x,y).

Theorem . The period of the Jordan-Fibonacci orbitJO(Qn;x,y)is P()·n–= ·n–.

Proof We prove this by direct calculation. We first note that|x|= n–_{,|y|=  andxy=} yx–_{. Since}

a_n()={, , , , , , , , , . . .},

it is clear thatP_{() = . From Definition ., it is easy to see that the Jordan-Fibonacci}

orbitJO(Qn;x,y) is as follows:

b_n+= b_n– b_n_+– b_n+ b_n+

forn≥, with initial conditions

Then we have the sequence

x,y,xn––y,xn–,e,xn–+,yx,

yx,x–,x–n––,x–,yx–,xn–+,xn–+, . . . .

So, the Jordan-Fibonacci orbitJO(Qn;x,y) can be said to form layers of length . Using the above, the sequence becomes

b_=x,b_=y,b_=xn––,b_=xn–, . . . ,

b_=x–,b_=x–y,b_ =xn–+y,b_=xn–+, . . . ,

b_i+=x–i+,b_i+=x–iy,b_i+=xi+n––,b_i+=xi+n–, . . . .

Hence, we need the smallesti∈Nsuch that i= n–_α

 and i= n–α. If we choose i= n–_{, we obtain}

b_·n–_+=x,b_·n–_+=y,b_·n–_+=x n–_–

,b_·n–_+=x n–

.

Since the elements succeeding b

·n–_+, b__·n–_+, b_·n–_+ andb_·n–_+ depend onx, y,

xn––_andxn–_{for their values, the cycle begins again with the (·}n–_{)nd element. Then}

we get that

PJO(Qn;x,y) =P()·n–= ·n–.

Thus it is verified thatPA(Q:x,y) = .

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics, Faculty of Science and Letters, Kafkas University, Kars, 36100, Turkey.}2_{Department of}

Actuarial Sciences, Faculty of Science, Atatürk University, Erzurum, 25240, Turkey.3_{Department of Mathematics,}

University Putra Malaysia, Serdang, Selangor 43400 UPM, Malaysia.

Acknowledgements

The authors express their sincere thanks to the referee for his/her careful reading and suggestions that helped to improve this paper.

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Received: 9 February 2017 Accepted: 11 April 2017

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