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Discrete Dynamics in Nature and Society
Volume 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 π‘š
Enci Gültekin1 and Yasemin TaGyurdu2
1
2
Department of Mathematics, Faculty of Science, Ataturk University, 25240 Erzurum, Turkey
Department of Mathematics, Faculty of Science and Letters, Erzincan University, 24000 Erzincan, Turkey
Correspondence should be addressed to IΜ‡nci Gültekin; igultekin@atauni.edu.tr
Received 20 November 2012; Revised 21 January 2013; Accepted 23 January 2013
Academic Editor: Binggen Zhang
Copyright © 2013 IΜ‡. Gültekin and Y. TasΜ§yurdu. 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.
It is shown that the sequence obtained by reducing modulo π‘š coefficient and exponent of each Fibonacci polynomials term is
periodic. Also if 𝑝 is prime, then sequences of Fibonacci polynomial are compared with Wall numbers of Fibonacci sequences
according to modulo 𝑝. It is found that order of cyclic group generated with 𝑄2 matrix ( π‘₯1 01 ) 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 𝐹𝑛 are the terms of the sequence 0, 1, 1, 2, 3, 5, . . ., where
𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2 , 𝑛 ≥ 2, with the initial values 𝐹0 = 0 and
𝐹1 = 1. 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 𝐹𝑛 (π‘₯) studied
by Catalan are defined by the recurrence relation
𝐹𝑛 (π‘₯) = π‘₯𝐹𝑛−1 (π‘₯) + 𝐹𝑛−2 (π‘₯) ,
𝑛 ≥ 3,
(1)
where 𝐹1 (π‘₯) = 1, 𝐹2 (π‘₯) = π‘₯. The Fibonacci polynomials studied by Jocobstral are defined by
𝐽𝑛 (π‘₯) = 𝐽𝑛−1 (π‘₯) + π‘₯𝐽𝑛−2 (π‘₯) ,
𝑛 ≥ 3,
(2)
where 𝐽1 (π‘₯) = 𝐽2 (π‘₯) = 1. The Fibonacci polynomials studied
by P. F. Byrd are defined by
πœ‘π‘› (π‘₯) = 2π‘₯πœ‘π‘›−1 (π‘₯) + πœ‘π‘›−2 (π‘₯) ,
𝑛 ≥ 2,
(3)
where πœ‘0 (π‘₯) = 0, πœ‘1 (π‘₯) = 1. The Lucas polynomials 𝐿 𝑛 (π‘₯),
originally studied in 1970 by Bicknell and they are defined by
𝐿 𝑛 (π‘₯) = π‘₯𝐿 𝑛−1 (π‘₯) + 𝐿 𝑛−2 (π‘₯) ,
𝑛 ≥ 2,
(4)
where 𝐿 0 (π‘₯) = 2, 𝐿 1 (π‘₯) = π‘₯ [1].
Hoggatt and Bicknell introduced a generalized Fibonacci
polynomials and their relationship to diagonals of Pascal’s
triangle [2]. Also after investigating the generalized 𝑄-matrix,
Ivie introduced a special case [3]. Nalli and Haukkanen
introduced β„Ž(π‘₯)-Fibonacci polynomials that generalize both
Catalan’s Fibonacci polynomials and Byrd’s Fibonacci Polynomials and the π‘˜-Fibonacci number. Also they provided
properties for these β„Ž(π‘₯)-Fibonacci polynomials where β„Ž(π‘₯)
is a polynomial with real coefficients [1].
Definition 1. The Fibonacci polynomials are defined by the
recurrence relation
0,
if 𝑛 = 0,
{
{
𝐹𝑛 (π‘₯) = {1,
if 𝑛 = 1,
{
+
𝐹
,
if
𝑛 ≥ 2,
π‘₯𝐹
(π‘₯)
(π‘₯)
𝑛−2
{ 𝑛−1
(5)
that the Fibonacci polynomials are generated by a matrix 𝑄2 ,
π‘₯ 1
𝑄2 = (
),
1 0
𝐹𝑛+1 (π‘₯) 𝐹𝑛 (π‘₯)
)
𝑄2𝑛 = (
𝐹𝑛 (π‘₯) 𝐹𝑛−1 (π‘₯)
(6)
2
Discrete Dynamics in Nature and Society
Table 1
Coefficient array
Fibonacci polynomials
𝐹0 (π‘₯) = 0
0
𝐹1 (π‘₯) = 1
1
𝐹2 (π‘₯) = π‘₯
By definition of the generalized Fibonacci polynomials we
have that 𝐹𝑖 (π‘₯)π‘š = π‘₯𝐹𝑖−1 (π‘₯)π‘š + 𝐹𝑖−2 (π‘₯)π‘š and 𝐹𝑗 (π‘₯)π‘š =
π‘₯𝐹𝑗−1 (π‘₯)π‘š + 𝐹𝑗−2 (π‘₯)π‘š . Hence, 𝐹𝑖 (π‘₯)π‘š = 𝐹𝑗 (π‘₯)π‘š , and then it
follows that
𝐹𝑖−1 (π‘₯)π‘š = 𝐹𝑗−1 (π‘₯)π‘š ,
1
2
𝐹3 (π‘₯) = π‘₯ + 1
𝐹𝑖−2 (π‘₯)π‘š = 𝐹𝑗−2 (π‘₯)π‘š , . . . , 𝐹𝑖−𝑗 (π‘₯)π‘š = 𝐹𝑗−𝑗 (π‘₯)π‘š = 𝐹0 (π‘₯)π‘š
(9)
1
1
1
2
𝐹5 (π‘₯) = π‘₯ + 3π‘₯ + 1
1
3
1
𝐹6 (π‘₯) = π‘₯5 + 4π‘₯3 + 3π‘₯
1
4
3
𝐹7 (π‘₯) = π‘₯6 + 5π‘₯4 + 6π‘₯2 + 1
..
.
1
..
.
5
..
.
6
..
.
𝐹4 (π‘₯) = π‘₯3 + 2π‘₯
4
2
which implies that the {𝐹(π‘₯)π‘š } is a periodic sequence.
1
..
.
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].
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 π‘Ž, 𝑏, 𝑐,
𝑑, 𝑒, 𝑏, 𝑐, 𝑑, 𝑒, 𝑏, 𝑐, 𝑑, 𝑒, . . ., is periodic after the initial element
π‘Ž and has period 4. A sequence is simply periodic with
period π‘˜ if the first π‘˜ elements in the sequence form a
repeating subsequence. For example, the sequence π‘Ž, 𝑏, 𝑐, 𝑑,
π‘Ž, 𝑏, 𝑐, 𝑑, π‘Ž, 𝑏, 𝑐, 𝑑, . . ., is simply periodic with period 4 [4]. The
minimum period length of (𝐹𝑖 mod 𝑛)∞
𝑖=−∞ sequence is stated
by π‘˜(𝑛) and is named Wall number of 𝑛 [5].
Example 4. For π‘š = 2, {𝐹(π‘₯)2 } sequence is 𝐹0 (π‘₯)2 = 0,
𝐹1 (π‘₯)2 = 1, 𝐹2 (π‘₯)2 = π‘₯, 𝐹3(π‘₯)2 = π‘₯2 + 1 = π‘₯0 + 1 = 2 = 0,
𝐹4 (π‘₯)2 = 0π‘₯ + π‘₯ = π‘₯, 𝐹5 (π‘₯)2 = π‘₯2 + 0 = π‘₯0 + 0 = 1,
𝐹6 (π‘₯)2 = π‘₯ + π‘₯ = 2π‘₯ = 0, 𝐹7 (π‘₯)2 = 0π‘₯ + 1 = 1. We have
{𝐹(π‘₯)2 } = {0, 1, π‘₯, 0, π‘₯, 1, 0, 1, . . .}, and then repeat. So, we get
β„ŽπΉ(π‘₯)2 = 6.
Given a matrix 𝐴 = (β„Žπ‘–π‘— (π‘₯)) where β„Žπ‘–π‘— (π‘₯)’s being polynomials with real coefficients, 𝐴(mod π‘š) means that every entry
of 𝐴 is modulo π‘š, that is, 𝐴(mod π‘š) = (β„Žπ‘–π‘— (π‘₯)(mod π‘š)).
Let βŸ¨π‘„2 βŸ©π‘š = {𝑄2𝑖 (mod π‘š) | 𝑖 ≥ 0} be a cyclic group and
|βŸ¨π‘„2 βŸ©π‘š | denote the order of βŸ¨π‘„2 βŸ©π‘š where 𝑄2𝑖 (mod π‘š) is
reduction coefficient and exponent of each polynomial in 𝑄2𝑖
matrix modulo π‘š.
Theorem 5. One has β„ŽπΉ(π‘₯)π‘š = |βŸ¨π‘„2 βŸ©π‘š |.
Proof. Proof is completed if it is that β„ŽπΉ(π‘₯)π‘š is divisible by
|βŸ¨π‘„2 βŸ©π‘š | and that |βŸ¨π‘„2 βŸ©π‘š | is divisible by β„ŽπΉ(π‘₯)π‘š . Fibonacci
polynomials are generated by a matrix 𝑄2 ,
π‘₯ 1
𝑄2 = (
),
1 0
Theorem 2. π‘˜(𝑛) is an even number for 𝑛 ≥ 3 [5].
2. The Generalized Sequence of
Fibonacci Polynomials Modulo π‘š
Reducing the generalized sequence of coefficient and exponent of each Fibonacci polynomials term by a modulus π‘š, we
can get a repeating sequence, denoted by
{𝐹(π‘₯)π‘š } = {𝐹0 (π‘₯)π‘š , 𝐹1 (π‘₯)π‘š , . . . , 𝐹𝑛 (π‘₯)π‘š , . . .} ,
(7)
where 𝐹𝑖 (π‘₯)π‘š = 𝐹𝑛 (π‘₯)(mod π‘š). Let β„ŽπΉ(π‘₯)π‘š denote the smallest period of {𝐹(π‘₯)π‘š }, called the period of the generalized
Fibonacci polynomials modulo π‘š.
Theorem 3. {𝐹(π‘₯)π‘š } is a periodic sequence.
𝐹𝑖+2 (π‘₯)π‘š = 𝐹𝑗+2 (π‘₯)π‘š , . . . , 𝐹𝑖+π‘˜ (π‘₯)π‘š = 𝐹𝑗+π‘˜ (π‘₯)π‘š .
Theorem 6. β„ŽπΉ(π‘₯)𝑝 = π‘π‘˜(𝑝) where 𝑝 is a prime number.
Proof. It is completed if it is that β„ŽπΉ(π‘₯)𝑝 is divisible by π‘π‘˜(𝑝)
and that π‘π‘˜(𝑝) divisible by β„ŽπΉ(π‘₯)𝑝 . From Theorem 5 𝑄2𝑛 =
𝑝
𝐹 (π‘₯) 𝐹𝑛 (π‘₯)
), 𝑄2β„ŽπΉ(π‘₯) = 𝐼(mod 𝑝) for 𝑄2 = ( π‘₯1 10 ). Also,
( 𝐹𝑛+1
𝑛 (π‘₯) 𝐹𝑛−1 (π‘₯)
π‘π‘˜(𝑝)
𝐹𝑖+1 (π‘₯)π‘š = 𝐹𝑗+1 (π‘₯)π‘š ,
(8)
(10)
Thus, it is clear that |βŸ¨π‘„2 βŸ©π‘š | is divisible by β„ŽπΉ(π‘₯)π‘š . Then
we need only to prove that β„ŽπΉ(π‘₯)π‘š is divisible by |βŸ¨π‘„2 βŸ©π‘š |.
𝐹 (π‘₯) 𝐹 (π‘₯)
Let β„ŽπΉ(π‘₯)π‘š = 𝑑. It is seen that 𝑄2𝑑 = ( 𝐹𝑑+1𝑑 (π‘₯) 𝐹𝑑−1𝑑 (π‘₯) ). Hence
𝑄2𝑑 = 𝐼(mod π‘š). We get that |βŸ¨π‘„2 βŸ©π‘š | is divisible by 𝑑. That
is, β„ŽπΉ(π‘₯)π‘š is divisible by |βŸ¨π‘„2 βŸ©π‘š |. So, we get β„ŽπΉ(π‘₯)π‘š =
|βŸ¨π‘„2 βŸ©π‘š |.
𝑄2
Proof. Let 𝑆2 = {(π‘₯1 , π‘₯2 ) : 1 ≤ π‘₯𝑖 ≤ 2} where π‘₯𝑖 is reduction
coefficient and exponent of each term in 𝐹𝑛 (π‘₯) polynomials
2
modulo π‘š. Then, we have |𝑆2 | = (π‘šπ‘š ) being finite, that is,
for any 𝑖 > 𝑗, there exist natural numbers 𝑖 and 𝑗
𝐹𝑛+1 (π‘₯) 𝐹𝑛 (π‘₯)
).
𝑄2𝑛 = (
𝐹𝑛 (π‘₯) 𝐹𝑛−1 (π‘₯)
=(
πΉπ‘π‘˜(𝑝)+1 (π‘₯) πΉπ‘π‘˜(𝑝) (π‘₯)
πΉπ‘π‘˜(𝑝) (π‘₯) πΉπ‘π‘˜(𝑝)−1 (π‘₯) ).
π‘π‘˜(𝑝)
So, we get 𝑄2
= 𝐼(mod 𝑝).
𝑝
Thus π‘π‘˜(𝑝) is divisible by β„ŽπΉ(π‘₯) . Moreover π‘π‘˜(𝑝) is divisible
by β„ŽπΉ(π‘₯)𝑝 . Since |βŸ¨π‘„2 βŸ©π‘ | = β„ŽπΉ(π‘₯)𝑝 , β„ŽπΉ(π‘₯)𝑝 is divisible by
π‘π‘˜(𝑝). Therefore β„ŽπΉ(π‘₯)𝑝 = π‘π‘˜(𝑝).
Theorem 7. β„ŽπΉ(π‘₯)𝑝 is an even number where 𝑝 is a prime
number.
Proof. It has been shown that β„ŽπΉ(π‘₯)𝑝 = π‘π‘˜(𝑝) in Theorem 6.
If it is stated that π‘π‘˜(𝑝) is an even number then proof is
Discrete Dynamics in Nature and Society
Table 2: Periods of the sequence of Fibonacci polynomials modulo
𝑝.
𝑝
2
7
37
103
181
241
373
653
853
π‘˜(𝑝)
3
16
76
208
90
240
748
1308
1708
β„ŽπΉ(π‘₯)𝑝
6
112
2812
21424
16290
57840
279004
854124
1456924
Result
β„ŽπΉ(π‘₯)2 = 2π‘˜ (2)
β„ŽπΉ(π‘₯)7 = 7π‘˜ (7)
β„ŽπΉ(π‘₯)37 = 37π‘˜ (37)
β„ŽπΉ(π‘₯)103 = 103π‘˜ (103)
β„ŽπΉ(π‘₯)181 = 181π‘˜ (181)
β„ŽπΉ(π‘₯)241 = 241π‘˜ (241)
β„ŽπΉ(π‘₯)373 = 373π‘˜ (373)
β„ŽπΉ(π‘₯)653 = 653π‘˜ (653)
β„ŽπΉ(π‘₯)853 = 853π‘˜ (853)
completed. By Theorem 2, π‘˜(𝑝) is an even number and 𝑝 is
an even number for 𝑝 ≥ 3. Hence π‘π‘˜(𝑝) is always an even
number. That is, β„ŽπΉ(π‘₯)𝑝 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 π‘˜(𝑝).
References
[1] A. Nalli and P. Haukkanen, “On generalized Fibonacci and
Lucas polynomials,” Chaos, Solitons and Fractals, vol. 42, no. 5,
pp. 3179–3186, 2009.
[2] 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.
[3] J. Ivie, “A general 𝑄-matrix,” The Fibonacci Quarterly, vol. 10, no.
3, pp. 255–261, 1972.
[4] S. W. Knox, “Fibonacci sequences in finite groups,” The
Fibonacci Quarterly, vol. 30, no. 2, pp. 116–120, 1992.
[5] D. D. Wall, “Fibonacci series modulo π‘š,” The American Mathematical Monthly, vol. 67, pp. 525–532, 1960.
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