On Period of the Sequence of Fibonacci Polynomials Modulo m E

F 1 = 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 F n (x) studied by Catalan are defined by the recurrence relation


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 0, 1, 1, 2, 3, 5, . .., where   =  −1 +  −2 ,  ≥ 2, with the initial values  0 = 0 and  1 = 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   () studied by Catalan are defined by the recurrence relation where  1 () = 1,  2 () = .The Fibonacci polynomials studied by Jocobstral are defined by where  1 () =  2 () = 1.The Fibonacci polynomials studied by P. F. Byrd are defined by where  0 () = 0,  1 () = 1.The Lucas polynomials   (), originally studied in 1970 by Bicknell and they are defined by where  0 () = 2,  1 () =  [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 ℎ()-Fibonacci polynomials that generalize both Catalan's Fibonacci polynomials and Byrd's Fibonacci Polynomials and the -Fibonacci number.Also they provided properties for these ℎ()-Fibonacci polynomials where ℎ() 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  2 , Table 1 Fibonacci polynomials Coefficient array 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 (  mod ) ∞ =−∞ sequence is stated by () and is named Wall number of  [5].
Theorem 7. ℎ()  is an even number where  is a prime number.
Proof.It has been shown that ℎ()  = () 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  ≥ 3. Hence () is always an even number.That is, ℎ()  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 .