Generalized commutative quaternions of the Fibonacci type

Quaternions are a four-dimensional hypercomplex number system discovered by Hamilton in 1843 and next intensively applied in mathematics, modern physics, computer graphics and other fields. After the discovery of quaternions, modified quaternions were also defined in such a way that commutative property in multiplication is possible. That number system called as commutative quaternions is intensively studied and used for example in signal processing. In this paper we define generalized commutative quaternions and next based on them we define and explore Fibonacci type generalized commutative quaternions.


Introduction
One of the problems in applying quaternions is their noncommutative structure. In other words, a different order of multiplication of quaternions results in different quaternions. That structure makes hard to conduct applications among other to engineering problems, see for example [22]. Modified quaternions were proposed by Serge in [26], so that commutative property in multiplication is possible. Quaternions and commutative quaternions belong to the class of hypercomplex numbers. Let N ! 0 be an integer. Hypercomplex numbers are defined by the expression (see [5,21,27]) x i e i where x i 2 R are called components and e i are called units. The element e 0 is the scalar or the real element and it may be identified with the real number 1. For i ¼ 1; . . .; N À 1 we have e i 6 2 R: Let x ¼ P NÀ1 i¼0 x i e i and y ¼ P NÀ1 i¼0 y i e i be any two hypercomplex numbers. We define equality, addition and multiplication by scalar in the following way: x ¼ y only if x i ¼ y i ; i ¼ 0; 1; . . .; N À 1 (equality) ðs Á x i Þe i ðmultiplication by scalar s 2 RÞ: The zero element is the element with all components equal to zero. From these definitions it follows that the addition satisfies commutative and associative properties and the zero element exists, see [6]. Important in defining the multiplication of hypercomplex numbers is that the product of hypercomplex numbers is still a hypercomplex number.
For real and complex numbers the product operation satisfies the following properties: distributive with respect to the sum, associative, commutative, does not have divisors of zero. It can be shown (see [25]) that, only the real and complex numbers can satisfy these four properties all together. As far as other systems are concerned, if they satisfy the first two properties they can satisfy just one of the last two.
Analogously as for quaternions the class (family) of hypercomplex numbers can be partitioned into two subclasses (subfamilies) non-commutative hypercomplex system and commutative hypercomplex system.
A generalized noncommutative quaternion x is a vector in 4-dimensional vectorspace of the form x ¼ x 0 þ x 1 e 1 þ x 2 e 2 þ x 3 e 3 , where quaternionic units e 1 ; e 2 ; e 3 satisfy the equalities e 2 1 ¼ Àa; e 2 2 ¼ Àb; e 2 3 ¼ Àab; e 1 e 2 ¼ Àe 2 e 1 ¼ e 3 ; e 2 e 3 ¼ Àe 3 e 2 ¼ be 1 and e 3 e 1 ¼ Àe where a; b 2 R: The class of all generalized noncommutative quaternions we denote by H n ab : For special values of a and b we obtain well-known subclasses of noncommutative quaternions Algebraic properties of generalized quaternions were described in [16]. Motivated by above we introduce generalized commutative quaternions as follows.
A generalized commutative quaternion x is a vector of the form where quaternionic units e 1 ; e 2 ; e 3 satisfy the equalities where a; b 2 R: The family of all generalized commutative quaternions we denote by H c ab : For special a and b there are the following subclasses of commutative quaternions Note that the above special cases are well-known and studied in the literature. It suffices to mention that in [6] Catoni et al. studied algebraic properties of three types of commutative quaternions i.e. elliptic, parabolic and hyperbolic quaternions. Although the theory of commutative quaternions is becoming important in recent years and has interesting applications in distinct areas of mathematics and modern physics, the study of commutative quaternions is focus mainly on elliptic quaternions, see for example [17][18][19][20]. Tessarines were introduced by Cocle in 1848 and next that theory was developed in the sequences of articles [7][8][9][10]. Bicomplex numbers were discovered by Serge, see [26] and recently studied by Rochon in [24]. Hyperbolic complex numbers also named as complex numbers with hyperbolic coefficients were investigated in [1]. Bihyperbolic numbers are one of the extension of hyperbolic numbers, see [4,23].
In recent years the theory of hypercomplex numbers is complemented by their connections with Fibonacci sequences, for more details see the book [27].
In this paper we define generalized commutative quaternions of the Fibonacci type.

Generalized commutative quaternions of the Fibonacci type
Let n ! 0 be an integer. The nth Fibonacci number F n is a number defined recursively by the second-order linear recurrence relation F n ¼ F nÀ1 þ F nÀ2 ; for n ! 2 with F 0 ¼ 0, F 1 ¼ 1: The Fibonacci numbers are generalized in different ways, see for example the list in [3] and [27]. Consider the well-known generalization of Fibonacci numbers given by Horadam in [14].
Let p, q, n be integers. For n ! 0 Horadam defined the numbers W n ¼ W n ðW 0 ; W 1 ; p; qÞ by the recursive equation for n ! 2 with fixed real numbers W 0 ; W 1 : For the historical reasons these numbers were later called the Horadam numbers. Let t 1 ; t 2 be the two distinct real roots of the equation Then, the Binet type formula for the Horadam numbers has the form where For special W 0 ; W 1 ; p; q the equation (3) defines selected numbers of the Fibonacci type: the Fibonacci numbers F n ¼ W n ð0; 1; 1; À1Þ, the Lucas numbers L n ¼ W n ð2; 1; 1; À1Þ, the Pell numbers P n ¼ W n ð0; 1; 2; À1Þ, the Pell-Lucas numbers Q n ¼ W n ð2; 2; 2; À1Þ, the Jacobsthal numbers J n ¼ W n ð0; 1; 1; À2Þ, the Jacobsthal-Lucas numbers j n ¼ W n ð2; 1; 1; À2Þ, the balancing numbers B n ¼ W n ð0; 1; 6; 1Þ, the Lucas-balancing numbers C n ¼ W n ð1; 3; 6; 1Þ etc.
Let n ! 0 be an integer. The nth generalized commutative Horadam quaternion gcH n is defined as where W n is the nth Horadam number and e 1 ; e 2 ; e 3 are units satisfy (1) and (2).
Theorem 1 (Binet type formula for generalized commutative Horadam quaternions) Let n ! 0 be an integer. Then where t 1 , t 2 , A and B are given by (5).
Proof Using (6) and (4) we have and the result follows. h For simplicity of notation let Then we can write (7) as Theorem 2 (Catalan type identity for generalized commutative Horadam quaternions) Let n ! 0; r ! 0 be integers such that n ! r: Then where t 1 , t 2 , A, B andt 1 ,t 2 are given by (5) and (8), respectively.
Proof By formula (7) which ends the proof. h Note that for r ¼ 1 we get the Cassini type identity for generalized commutative Horadam quaternions.