The quaternion-type cyclic-Fibonacci sequences in groups

: In this paper, we define the six different quaternion-type cyclic-Fibonacci sequences and present some properties, such as, the Cassini formula and generating function. Then, we study quaternion-type cyclic-Fibonacci sequences modulo 𝑚 . Also we present the relationships between the lengths of periods of the quaternion-type cyclic-Fibonacci sequences of the first, second, third, fourth, fifth and sixth kinds modulo 𝑚 and the generating matrices of these sequences. Finally, we introduce the quaternion-type cyclic-Fibonacci sequences in finite groups. We calculate the lengths of periods for these sequences of the generalized quaternion groups and obtain quaternion-type cyclic-Fibonacci orbits of the quaternion groups 𝑄 8 and 𝑄 16 as applications of the results.

It is well known that the Fibonacci sequence { } is defined by the following homogeneous linear recurrence relation: for ≥ 2, where 0 = 0 and 1 = 1. In [22], it can be obtained miscellaneous properties involving Fibonacci numbers. The initial work began with Fibonacci sequences in algebraic structures that Wall [28] investigated in cyclic groups. Number theoretic properties such as these get from homogeneous linear recurrence relations relevant to this subject have been researched recently by many authors; see for example, [2-15, 17-21, 23, 25, 26, 29]. In [1], the author studied the complex-type Pell -numbers modulo and get the periods and the ranks of the complex-type Pell -numbers modulo . Deveci and Shannon [11] extended the theory to the quaternions. Lü and Wang demonstrated that the -step Fibonacci sequence modulo is simply periodic [24]. After a given point, a sequence is considered periodic if all it consists of is repeated iterations of a fixed subsequence. The number of elements in the shortest repeating subsequence determines the period of sequence. As an illustration, the sequence , , , ℎ, , , , ℎ, , , , ℎ, , . . . is periodic and has a period of 4 following the first element . If the first components of a sequence form a repeating subsequence, the sequence is simply periodic with period . The sequence , , , ℎ, , , , , ℎ, , , , , ℎ, , . . . , for instance, is merely periodic with period 5.
In Section 2, we define the six different quaternion-type cyclic-Fibonacci sequences and then present some properties, such as, the Cassini formulas, generating function. Also, we get the relationship between the Fibonacci sequence and the first three quaternion-type cyclic-Fibonacci numbers. In Section 3, we study quaternion-type cyclic-Fibonacci sequences modulo and then, we give the relationships between the lengths of periods of the quaternion-type cyclic-Fibonacci sequences of the first, second, third, fourth, fifth and sixth kind modulo and the generating matrices of these sequences. In Section 4, we introduce the quaternion-type cyclic-Fibonacci sequences in groups. After this, we calculate the quaternion Fibonacci lengths of generalized quaternion groups. Finally, we give a specific example for sequences of quaternion groups 8 and 16 .

The quaternion-type cyclic-Fibonacci sequences
In this section, we will introduce six different quaternion-type cyclic-Fibonacci sequences for any positive integer number ≥ 2. Then, we will present miscellaneous properties of these sequences.
Since the multiplication of quaternions is not commutative, the above properties are given considering multiplicative order. Therefore, it is easy to see that In order to easy in our operations, we define ( ) as follows: where ∈ Z + . We will give relation these sequences to the well-known classic Fibonacci sequence where = 1, 2, 3 and ( ) is as defined in the Equation (2.1). Now, we introduce matrices for the quaternion-type cyclic-Fibonacci sequences, similar to the -matrix for classic Fibonacci sequence. We can write for these sequences ]︃ for = 4, 5, 6.
( ). The proof will only be done for the case = 4, the others are done similarly. By Definition 2.1, we get And then, since ( + ) 4 3 Similarly, we can write And then, since ( + ) 4 Similarly, we have And then, since ( + ) 4 From the Equations (2.11), (2.12) and (2.13), we obtain as required.
In the following theorem, we develop the generating functions for the quaternion-type cyclic-Fibonacci sequences.
Proof. ( ). Assume that ( ) is the generating function of the { } for = 1, 2, 3. Then we have From Lemma 2.1, we obtain Now the rearrangement of the equation implies that which equals to the left-hand sides in the Theorem.
( ). The proof can be done similarly to ( ).

The quaternion-type cyclic-Fibonacci sequence modulo
In this section, we study quaternion-type cyclic-Fibonacci sequences modulo . Then, we give the relationships between the lengths of periods of the quaternion-type cyclic-Fibonacci sequences of the first, second, third, fourth, fifth and sixth kind modulo and the generating matrices of these sequences. Let denote the -th member of the Fibonacci sequences 0 = , 1 = , +1 = + −1 ( ≥ 1).

Theorem 3.1. (Wall, [28]) ( mod ) forms a simply periodic sequence. That is, the sequence is periodic and repeats by returning to its starting values.
The length of the period of the ordinary Fibonacci sequence { } modulo was denoted by ( ).
If we take the least nonnegative residues and decrease the first, second, third, fourth, fifth, and sixth kinds of quaternion-type cyclic-Fibonacci sequences modulo , we obtain the following recurrence sequences: for every integer 1 ≤ ≤ 6, where ( ) is used to mean the -th element of the -th quaternion-type cyclic-Fibonacci sequence when read modulo . We observe here that the recurrence relations in the sequences { ( )} and { } are the same. Suppose that the cardinality of the set is denoted by the notation | |. Since the set is finite, there are | | distinct 2-tuples of the quaternion-type cyclic-Fibonacci sequences of the first kind { 1 } modulo . Thus, it is clear that at least one of these 2-tuples appears twice in the sequence . .. So, it is easy to see that the subsequence following this 2-tuple repeats; that is, { 1 ( )} is a periodic sequence and the length of its period must be divisible by 3.

The quaternion-type cyclic-Fibonacci sequence in groups
In this section, we will define six different quaternion-type cyclic-Fibonacci sequences in finite groups. Subsequently, we will examine the quaternion-type cyclic-Fibonacci orbits of the first, second, third, fourth, fifth and sixth kinds of the generalized quaternion group. Finally, we will give a specific example for sequences of quaternion groups 8 and 16 .
Let be a 2-generator group and let We call ( 1 , 2 ) a generating pair for .

Conclusion
In this paper, we defined the quaternion-type cyclic-Fibonacci sequences and then we obtained the relationships among the elements of these sequences and the generating matrices of these sequences. Also, we gave the Cassini formula, generating functions of the quaternion-type cyclic-Fibonacci sequences. Then, we studied the quaternion-type cyclic-Fibonacci sequences modulo . Furthermore, we got the cyclic groups generated by reducing the multiplicative orders of the generating matrices and the auxiliary equations of these sequences modulo and then, we investigated the orders of these cyclic groups. Moreover, using the terms of 2-generator groups which is called the quaternion-type cyclic-Fibonacci orbit, we redefined the quaternion-type cyclic-Fibonacci sequences. Also, these sequences in finite groups were examined in detail. With this study, we will gain a new perspective to the Fibonacci quaternions in the literature.