On Period of the Sequence of Fibonacci Polynomials Modulo
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.
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 , .
Hoggatt and Bicknell introduced a generalized Fibonacci polynomials and their relationship to diagonals of Pascal’s triangle . Also after investigating the generalized -matrix, Ivie introduced a special case . 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 .
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 .
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 . The minimum period length of sequence is stated by and is named Wall number of .
Theorem 2. is an even number for .
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 .