On the 2k-step Jordan-Fibonacci sequence - Advances in Difference

Deveci et al. Advances in Difference Equations (2017) 2017:121 DOI 10.1186/s13662-017-1178-2

RESEARCH

Open Access

On the k-step Jordan-Fibonacci sequence Omur Deveci1 , Sait Ta¸s2 and A Kılıçman3* *

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 modulo m. Also, we obtain the cyclic groups from the multiplicative orders of the generating matrix of the 2k-step Jordan-Fibonacci sequence when read modulo m, and we give the relationships among the orders of the cyclic groups obtained and the periods of the 2k-step Jordan-Fibonacci sequence modulo m. 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 group Q2n as 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 combination of the preceding k terms:

an+k = c an + c an+ + · · · + ck– an+k– ,

where c , 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 matrix A be defined by ⎡

 ⎢ ⎢ ⎢ ⎢ ⎢ A = [aij ]k×k = ⎢ ⎢. ⎢ .. ⎢ ⎢ ⎣ c

 

 

 .. .  c

 .. .  c

... ... .. .

...

   .. .  ck–

⎤  ⎥  ⎥ ⎥ ⎥  ⎥ ⎥, .. ⎥ . ⎥ ⎥ ⎥  ⎦ ck–

© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Deveci et al. Advances in Difference Equations (2017) 2017:121

then ⎡

⎤ ⎡ ⎤ a an ⎢a ⎥ ⎢a ⎥ ⎢  ⎥ ⎢ n+ ⎥ n⎢ ⎥ ⎢ A ⎢ . ⎥=⎢ . ⎥ ⎥ ⎣ .. ⎦ ⎣ .. ⎦ ak– an+k– for n ≥ . k = , Fkk =  The k-step Fibonacci sequence {Fnk } is defined by initial values Fk = · · · = Fk– and the recurrence relation k Fn+k =

k– 

k Fn+i

i=

for n ≥ . It is clear that the characteristic polynomial of the k-step Fibonacci sequence is as follows: Pk (x) = xk – xk– – · · · – x – . A square matrix of the form ⎡

λ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢. ⎢ .. ⎢ ⎢ ⎣ 

 λ  .. .  

  λ .. .  

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

   .. . λ 

⎤  ⎥ ⎥ ⎥ ⎥ ⎥ .. ⎥ .⎥ ⎥ ⎥ ⎦ λ

is called a Jordan block. The Jordan block of order k is denoted by Jk (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 [] extended 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 matrix J with the aid of the characteristic polynomial of the k-step Fibonacci sequence, and then we define the k-step JordanFibonacci sequence by using the Jordan-Fibonacci matrix FJk of order k which is produced from the matrix J. Then we study the k-step Jordan-Fibonacci sequence modulo m, and we obtain the cyclic groups from the multiplicative orders of the Jordan-Fibonacci matrix FJk of order k such that the elements of the matrix FJk when read modulo m. Also, we derive the relationships among the orders of the cyclic groups obtained and the periods of the k-step Jordan-Fibonacci sequence modulo m. Furthermore, we redefine the kstep Jordan-Fibonacci sequence by means of the elements of the groups which have two or more generators, and then we examine this sequence in the finite groups. Finally, we

Page 2 of 9

Deveci et al. Advances in Difference Equations (2017) 2017:121

Page 3 of 9

obtain the period of the k-step Jordan-Fibonacci sequence in the generalized quaternion group Qn as applications of the results produced.

2 Main results and proofs We consider the (k)×(k) Jordan-type matrix J which is defined by using the characteristic polynomial of the k-step Fibonacci sequence: ⎡

 ⎢ ⎢ ⎢. ⎢. ⎣. 

– 

– – .. .

... ... .. .

– –

 –

 

... ...

...





–

–

–

...

⎤  ⎥ ⎥ ⎥. ⎥ ⎦ –

Now we define the Jordan-Fibonacci matrix FJk = [fij ]k×k by using the last row of the matrix J as follows: k th ⎡

 ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 

↓

  

...  



..

 ... ...

–

...  

 

 

 

.

... ...

⎤ – ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 

For example, ⎡

 ⎢ ⎢ FJ = ⎢ ⎣ 

   

–   

⎤ – ⎥ ⎥ ⎥. ⎦ 

Next, we define the k-step Jordan-Fibonacci sequence with the aid of the matrix FJk as follows: akn+k = akn+k– + akn+k – akn+k– – · · · – akn

()

for n ≥ , with initial conditions ak = · · · = akk– =  and akk = . By mathematical induction on α, we may write ⎡  aα+   α ⎢ ⎢a FJ = ⎢ α+ ⎣aα+ aα+

aα+ – aα+ aα+ – aα+ aα+ – aα+ aα+ – aα+

–aα+ – aα+ –aα+ – aα+ –aα+ – aα aα – aα–

⎤ –aα+ –aα+ ⎥ ⎥ ⎥ –aα+ ⎦ –aα

()

Deveci et al. Advances in Difference Equations (2017) 2017:121

Page 4 of 9

and ⎡



α FJk

=

ak ⎢ α+k ⎢ k ⎢ aα+k– ⎢ ⎢ k ⎢a ⎢ α+k– ⎢ ⎢ . ⎢ . ⎢ ⎢ . ⎣ akα+

ak – ak α+k+ α+k ak – ak α+k α+k– ak – ak α+k– α+k–

ak – ak α+k+ α+k+ ak – ak α+k+ α+k ak – ak α+k α+k–

. . .

. . .

akα+ – akα+

akα+ – akα+

... ... ...

ak – ak α+k– α+k– ak – ak α+k– α+k– ak – ak α+k– α+k– . . .

...

ak – ak α+k α+k–

M

⎤ –ak α+k– ⎥ ⎥ –ak α+k– ⎥ ⎥ ⎥ ⎥ –ak α+k– ⎥ ⎥ ⎥ . ⎥ . ⎥ ⎥ . ⎦ –akα

()

for k ≥ , where M is a (k) × (k) matrix as follows: ⎡

–akα+k– – akα+k– – · · · – akα+k ⎢ k ⎢ –aα+k– – akα+k– – · · · – akα+k– ⎢ k k ⎢–ak α+k– – aα+k– – · · · – aα+k– M=⎢ ⎢ ⎢ . ⎢ . . ⎣

–akα+k– – akα+k– – · · · – akα+k+ –akα+k– – akα+k– – · · · – akα+k –akα+k– – akα+k– – · · · – akα+k–

–akα – akα– – · · · – akα–k+

...

–akα+k– – akα+k–

...

–akα+k– –akα+k–

...

. . .

. . . –akα – akα– – · · · – akα–k+

...

⎤

⎥ – akα+k– ⎥ ⎥ k – aα+k– ⎥ ⎥. ⎥ ⎥ ⎥ ⎦

–akα – akα–

Now we consider the k-step Jordan-Fibonacci sequence {akn } modulo m. If we reduce the sequence {akn } by a modulus m, then we get the repeating sequence, denoted by  k  an (m) = ak (m), ak (m), . . . , aki (m), . . . , where we denote aki (mod m) by aki (m). It has the same recurrence relation as in (). A sequence is said to be periodic if, 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 first n elements in the sequence form a repeating subsequence, then this sequence is simply periodic and its period is n. Theorem . {akn (m)} forms a simply periodic sequence. Proof Let X = {(x , x , . . . , xk )|xi ’s be integers such that  ≤ xi ≤ m – }. Since there are mk distinct k-tuples of elements of Zm , at least one of the k-tuples appears twice in the sequence {akn (m)}. Therefore, the subsequence following this k-tuple repeats; hence, the sequence is periodic. Assume that u > v and aku+ (m) = akv+ (m), aku+ (m) = akv+ (m), . . . , aku+k (m) = akv+k (m), then u ≡ v (mod k). By (), we may write akn = –akn+k + akn+k– + akn+k – akn+k– – · · · – akn+ . Then we obtain aku (m) = akv (m), aku– (m) = akv– (m), . . . , aku–v+ (m) = ak (m). Thus it is verified that the sequence is simply periodic. The period of the sequence {akn (m)} is denoted by Pk (m).



Deveci et al. Advances in Difference Equations (2017) 2017:121

Given an integer matrix A = [aij ], A (mod m) means that all entries of A are modulo m, that is,

A (mod m) = aij (mod m) . Let us consider the set  Am = Ai (mod m)|i ≥  . If gcd(m, det A) = , then Am is a cyclic group; if gcd(m, det A) = , then Am is a semigroup. Let the notation |Am | denote the order of the set Am . Since det FJk = , |FJk m | is a cyclic group for every positive integer m. From () and (), it is easy to see that Pk (m) = |FJk m |. Now we give some properties of the period Pk (m) by the following theorems. Theorem . Let t be a prime and suppose that u is the largest positive integer with Pk (t) = Pk (t u ). Then

Pk t v = t v–u · Pk (t) for every v ≥ u. In particular if Pk (t  ) = Pk (t), then Pk (t v ) = t v– · Pk (t). Proof Since Pk (m) = |FJk m |, we have a positive integer n such that

FJk

Pk (tn+ )

≡ I mod t n+ ,

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

FJk

Pk (tn+ )



≡ I mod t n ,

which implies that Pk (t n+ ) is divisible by Pk (t n ). Furthermore, if we denote

FJk

Pk (tn )

= I + fij(t) · t n ,

then by the binomial expansion, we may write

FJk

Pk (tn )·t

t = I + fij(t) · t n =

t   t i=

i

i

fij(t) · t n ≡ I mod t n+ .

This yields that Pk (t n+ ) divides Pk (t n ) · t. Then it is clear that



Pk t n+ = Pk t n or



Pk t n+ = Pk t n+ · t.

Page 5 of 9

Deveci et al. Advances in Difference Equations (2017) 2017:121

Page 6 of 9

It is easy to see that the latter holds if and only if there is fij(t) which is not divisible by t. Since u is the largest positive integer such that

Pk (t) = Pk t u ,



Pk t u = Pk t u+ , then there is fij(u+) which is not divisible by t. Therefore, we obtain



Pk t u+ = Pk t u+ , and so





Pk t n+ = Pk t n+ · t = Pk t n · t  . 

To complete the proof we may use an inductive method. Theorem . If m has the prime factorization m = r equals the least common multiple of Pk (ti i ) s. r

u

ri i= ti ,

where (u ≥ ), then Pk (m)

r

Proof Since Pk (ti i ) is a period of the sequence {akn (ti i )}, it is easy to see that the sequence r r {akn (ti i )} repeats only after blocks of length n · Pk (ti i ), (n ∈ N). Since also Pk (m) is a period r of the sequence {akn (m)}, the sequence {akn (ti i )} repeats after terms Pk (m) for all values i. r Then we get that Pk (m) is the form n · Pk (pi i ) for all values of i, and since any such number gives a period of Pk (m), we conclude that  

Pk (m) = lcm Pk tr , . . . , Pk turu .



Now we extend the k-step Jordan-Fibonacci sequence to k-generator groups such that k ≥ . Let G be a finite k-generator group and suppose that    X = (x , x , . . . , xk ) ∈ G × G × · · · × G | {x , x , . . . , x } = G .   k    k

We call (x , x , . . . , xk ) a generating k-tuple for G. Definition . For a k-tuple (x , x , . . . , xk ) ∈ X, we define the Jordan-Fibonacci orbit  JO(G; x , x , . . . , xk ) = bki by bk = x , . . . , bkk = xk , bkk+ = (x )– (x )– · · · (xk )– , bkk+ = · · · = bkk = (x )– (x )– · · · (xk )– (xk+ ) and

– k k

– bkn+k = bkn · · · bkn+k– bn+k bn+k– for n ≥ .

Deveci et al. Advances in Difference Equations (2017) 2017:121

Page 7 of 9

Theorem . If the group G is finite, then the Jordan-Fibonacci orbit JO(G; x , x , . . . , xk ) is simply periodic. Proof Let β be order of the group G, then it is clear that there are β k distinct k-tuples of elements of G. Thus it is verified that at least one of the k-tuples appears twice in the Jordan-Fibonacci orbit JO(G; x , x , . . . , xk ). Because of the repeating, the sequence is periodic. Since the Jordan-Fibonacci orbit JO(G; x , x , . . . , xk ) is periodic, there exist natural numbers m and l with l ≡ m (mod k) such that bkm+ = bkl+ , bkm+ = bkl+ , . . . , bkm+k = bkl+k . From the definition of the Jordan-Fibonacci orbit JO(G; x , x , . . . , xk ), we may write

–

– k k



– bkn = bkn+ · · · bkn+k– bn+k bn+k– bkn+k . Thus it is verified that bkm = bkl , bkm– = bkl– , . . . , bk = bkl–m+ . 

So the proof is complete.

The period of the Jordan-Fibonacci orbit JO(G; x , x , . . . , xk ) is denoted by PJO(G; x , x , . . . , xk ). It is well known that the generalized quaternion group Qn is defined by presentation   n– n– x, y|x = e, y = x , y– xy = x– , for n ≥ . We will now address the Jordan-Fibonacci orbit of the generalized quaternion group Qn for generating pair (x, y). Theorem . The period of the Jordan-Fibonacci orbit JO(Qn ; x, y) is P () · n– =  · n– . Proof We prove this by direct calculation. We first note that |x| = n– , |y| =  and xy = yx– . Since   an () = {, , , , , , , , , . . .}, it is clear that P () = . From Definition ., it is easy to see that the Jordan-Fibonacci orbit JO(Qn ; x, y) is as follows: –  –  

bn+ bn+ bn+ bn+ = bn for n ≥ , with initial conditions n– –

b = x, b = y, b = x

n–

, b = x

.

Deveci et al. Advances in Difference Equations (2017) 2017:121

Page 8 of 9

Then we have the sequence n– –

x, y, x

n–

y, x

n– –

yx, x– , x–

n– +

, e, x

, yx , n– +

, x– , yx– , x

n– +

, x

,....

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

b = x, b = y, b = x

n–

, b = x

b = x– , b = x– y, b = x

n– +

,..., n– +

y, b = x

n– –

bi+ = x–i+ , bi+ = x–i y, bi+ = xi+

,..., n–

, bi+ = xi+

,....

Hence, we need the smallest i ∈ N such that i = n– α and i = n– α . If we choose i = n– , we obtain n– –

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

n–

, b·n– + = x

.

Since the elements succeeding b·n– + , b·n– + , b·n– + and b·n– + depend on x, y, n– n– x – and x 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 that PA(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.

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 9 February 2017 Accepted: 11 April 2017 References 1. Kalman, D: Generalized Fibonacci numbers by matrix methods. Fibonacci Q. 20(1), 73-76 (1982) 2. Wall, DD: Fibonacci series modulo m. Am. Math. Mon. 67(6), 525-532 (1960) 3. Wilcox, HJ: Fibonacci sequences of period n in groups. Fibonacci Q. 24(4), 356-361 (1986) 4. Aydin, H, Smith, GC: Finite p-quotients of some cyclically presented groups. J. Lond. Math. Soc. 49(1), 83-92 (1994) 5. Campbell, CM, Doostie, H, Robertson, EF: Fibonacci length of generating pairs in groups. In: Bergum, GE, Philippou, AN, Horadam, AF (eds.) Applications of Fibonacci Numbers, vol. 3, pp. 27-35. Kluwer Academic, Dordrecht (1990)

Deveci et al. Advances in Difference Equations (2017) 2017:121

6. Campbell, CM, Campbell, PP: The Fibonacci length of certain centro-polyhedral groups. J. Appl. Math. Comput., Int. J. 19(1), 231-240 (2005) 7. Deveci, O, Karaduman, E: The Pell sequences in finite groups. Util. Math. 96, 263-276 (2015) 8. Doostie, H, Hashemi, M: Fibonacci lengths involving the Wall number k(n). J. Appl. Math. Comput. 20(1-2), 171-180 (2006) 9. Karaduman, E, Aydın, H: On the relationship between the recurrences in nilpotent groups and the binomial formula. Indian J. Pure Appl. Math. 35(9), 1093-1103 (2004) 10. Knox, SW: Fibonacci sequences in finite groups. Fibonacci Q. 30(2), 116-120 (1992) 11. Lü, K, Wang, J: k-Step Fibonacci sequence modulo m. Util. Math. 71, 169-177 (2006) 12. Ozkan, E, Aydin, H, Dikici, R: 3-step Fibonacci series modulo m. Appl. Math. Comput. 143(1), 165-172 (2003) 13. Tas, S, Deveci, O, Karaduman, E: The Fibonacci-Padovan sequences in finite groups. Maejo Int. J. Sci. Technol. 8(3), 279-287 (2014) 14. Tas, S, Karaduman, E: The Padovan sequences in finite groups. Chiang Mai J. Sci. 41(2), 456-462 (2014) 15. Deveci, O, Karaduman, E: The cyclic groups via the Pascal matrices and the generalized Pascal matrices. Linear Algebra Appl. 437(10), 2538-2545 (2012)

Page 9 of 9