Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 731482, 3 pages

http://dx.doi.org/10.1155/2013/731482

Research Article

On Period of the Sequence of Fibonacci Polynomials Modulo

İnci Gültekin¹ and Yasemin Taşyurdu²

¹Department of Mathematics, Faculty of Science, Ataturk University, 25240 Erzurum, Turkey

²Department of Mathematics, Faculty of Science and Letters, Erzincan University, 24000 Erzincan, Turkey

Received 20 November 2012; Revised 21 January 2013; Accepted 23 January 2013

Academic Editor: Binggen Zhang

Copyright © 2013 İnci Gültekin and Yasemin Taşyurdu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

It is shown that the sequence obtained by reducing modulo coefficient and exponent of each Fibonacci polynomials term is periodic. Also if is prime, then sequences of Fibonacci polynomial are compared with Wall numbers of Fibonacci sequences according to modulo . It is found that order of cyclic group generated with matrix is equal to the period of these sequences.

1. Introduction

In modern science there is a huge interest in the theory and application of the Fibonacci numbers. The Fibonacci numbers are the terms of the sequence , where ,  , with the initial values and . Generalized Fibonacci sequences have been intensively studied for many years and have become an interesting topic in Applied Mathematics. Fibonacci sequences and their related higher-order sequences are generally studied as sequence of integer. Polynomials can also be defined by Fibonacci-like recurrence relations. Such polynomials, called Fibonacci polynomials, were studied in 1883 by the Belgian mathematician Eugene Charles Catalan and the German mathematician E. Jacobsthal. The polynomials studied by Catalan are defined by the recurrence relation

where , . The Fibonacci polynomials studied by Jocobstral are defined by

where . The Fibonacci polynomials studied by P. F. Byrd are defined by

where , . The Lucas polynomials , originally studied in 1970 by Bicknell and they are defined by

where , [1].

Hoggatt and Bicknell introduced a generalized Fibonacci polynomials and their relationship to diagonals of Pascal’s triangle [2]. Also after investigating the generalized -matrix, Ivie introduced a special case [3]. Nalli and Haukkanen introduced -Fibonacci polynomials that generalize both Catalan’s Fibonacci polynomials and Byrd’s Fibonacci Polynomials and the -Fibonacci number. Also they provided properties for these -Fibonacci polynomials where is a polynomial with real coefficients [1].

Definition 1. The Fibonacci polynomials are defined by the recurrence relation

that the Fibonacci polynomials are generated by a matrix ,

can be verified quite easily by mathematical induction. The first few Fibonacci polynomials and the array of their coefficients are shown in Table 1 [2].

Table 1

Fibonacci polynomials
Coefficient array

0

1

1

1
1

1
2

1
3
1

1
4
3

1
5
6
1

A sequence is periodic if, after a certain point, it consists of only repetitions of a fixed subsequence. The number of elements in the repeating subsequence is called the period of the sequence. For example, the sequence , is periodic after the initial element and has period 4. A sequence is simply periodic with period if the first elements in the sequence form a repeating subsequence. For example, the sequence , is simply periodic with period 4 [4]. The minimum period length of sequence is stated by and is named Wall number of [5].

Theorem 2. is an even number for [5].

2. The Generalized Sequence of Fibonacci Polynomials Modulo

Reducing the generalized sequence of coefficient and exponent of each Fibonacci polynomials term by a modulus , we can get a repeating sequence, denoted by

where . Let denote the smallest period of , called the period of the generalized Fibonacci polynomials modulo .

Theorem 3.   is a periodic sequence.

Proof. Let where is reduction coefficient and exponent of each term in polynomials modulo . Then, we have being finite, that is, for any , there exist natural numbers and

By definition of the generalized Fibonacci polynomials we have that and . Hence, , and then it follows that

which implies that the is a periodic sequence.

Example 4. For , sequence is , , , ,  , , , . We have, and then repeat. So, we get .

Given a matrix where ’s being polynomials with real coefficients, means that every entry of is modulo , that is, . Let be a cyclic group and denote the order of where is reduction coefficient and exponent of each polynomial in matrix modulo .

Theorem 5. One has .

Proof. Proof is completed if it is that is divisible by and that is divisible by . Fibonacci polynomials are generated by a matrix ,

Thus, it is clear that is divisible by . Then we need only to prove that is divisible by . Let . It is seen that . Hence . We get that is divisible by . That is, is divisible by . So, we get .

Theorem 6. where is a prime number.

Proof. It is completed if it is that is divisible by and that divisible by . From Theorem 5  ,   for . Also, . So, we get . Thus is divisible by .  Moreover is divisible by . Since , is divisible by . Therefore .

Theorem 7. is an even number where is a prime number.

Proof. It has been shown that in Theorem 6. If it is stated that is an even number then proof is completed. By Theorem 2, is an even number and is an even number for . Hence is always an even number. That is, is an even number.

Table 2 shows some periods of sequence of coefficient and exponent of Fibonacci polynomials modulo, which is a prime number, by using .

Table 2: Periods of the sequence of Fibonacci polynomials modulo .

Result

2
3
6

7
16
112

37
76
2812

103
208
21424

181
90
16290

241
240
57840

373
748
279004

653
1308
854124

853
1708
1456924

References

A. Nalli and P. Haukkanen, “On generalized Fibonacci and Lucas polynomials,” Chaos, Solitons and Fractals, vol. 42, no. 5, pp. 3179–3186, 2009.

V. E. Hoggatt Jr. and M. Bicknell, “Generalized Fibonacci polynomials and Zeckendorf's theorem,” The Fibonacci Quarterly, vol. 11, no. 4, pp. 399–419, 1973.

J. Ivie, “A general -matrix,” The Fibonacci Quarterly, vol. 10, no. 3, pp. 255–261, 1972.

S. W. Knox, “Fibonacci sequences in finite groups,” The Fibonacci Quarterly, vol. 30, no. 2, pp. 116–120, 1992.

D. D. Wall, “Fibonacci series modulo ,” The American Mathematical Monthly, vol. 67, pp. 525–532, 1960.