Bicomplex Tetranacci and Tetranacci-Lucas Quaternions

In this paper, we introduce the bicomplex Tetranacci and Tetranacci-Lucas quaternions. Moreover, we present Binet’s formulas, generating functions, and the summation formulas for those bicomplex quaternions.


Introduction
In this paper, we define bicomplex Tetranacci and bicomplex Tetranacci-Lucas quaternions by combining bicomplex numbers and Tetranacci, Tetranacci-Lucas numbers and give some properties of them. Before giving their definition, we present some information on bicomplex numbers and also on Tetranacci and Tetranacci-Lucas numbers.
The bicomplex numbers (quaternions) are defined by the four bases elements 1, i, j, i j where i, j and i j satisfy the following properties: A bicomplex number can be expressed as follows: q = a 0 + ia 1 + ja 2 + i ja 3 = (a 0 + ia 1 ) + j(a 2 + ia 3 ) = z 0 + jz 1 , where a 0 , a 1 , a 2 , a 3 are real numbers and z 0 , z 1 are complex numbers. So the set of bicomplex number is BC = {z 0 + jz 1 : z 0 , z 1 ∈ C, j 2 = −1}.
We now give some basic informations on quaternions. Quaternions were formally invented by Irish mathematician W. R. Hamilton (1805-1865) as an extension to the complex numbers and for some background about this type of hypercomplex numbers we refer the works, for example, in [3], [6], [23]. The field H of quaternions is a four-dimensional non-commutative R-field generated by four base elements 1, i, j and k that satisfy the following rules: Briefly BC, the set of bicomplex numbers, has the following properties: • Quaternions and bicomplex numbers are generalizations of complex numbers, but one difference between them is that quaternions are non-commutative, whereas bicomplex numbers are commutative.
• Real quaternions are non-commutative, and don't have zero divisors and non-trivial idempotent elements. But bicomplex numbers are commutative, have zero divisors and non-trivial idempotent elements: • All above norms are isotropic. For example, for N i , we calculate N i (q) for q = 1 + i j as • BC is a real vector space with the addition of bicomplex numbers and the multiplication of a bicomplex number by a real scalar.
• BC forms a commutative ring with unity which contains C.
• BC forms a two-dimensional algebra over C, and since C is of dimension two over R, the bicomplex numbers are an algebra over R of dimension four.
• BC is a real associative algebra with the bicomplex number product ×.
For more details about these type of numbers (quaternions), we refer to, for example, the works [13], [18], among others.
The sequences {M n } n≥0 and {R n } n≥0 can be extended to negative subscripts by defining for n = 1, 2, 3, . . ., respectively. Therefore, recurrences (1.3) and (1.4) hold for all integer n. Table 2 presents the first few values of the Tetranacci and Tetranacci-Lucas numbers with positive and negative subscripts: It is well known that for all integers n, usual Tetranacci and Tetranacci-Lucas numbers can be expressed using Binet's formulas [24] or [10]), or (see for example [7]) and R n = α n + β n + γ n + δ n respectively, where α, β, γ and δ are the roots of the equation Note that the Binet form of a sequence satisfying (1.3) and (1.4) for non-negative integers is valid for all integers n. This result of Howard and Saidak [11] is even true in the case of higher-order recurrence relations.
The generating functions for Tetranacci sequence {M n } n≥0 and Tetranacci-Lucas sequence respectively.

The Bicomplex Tetranacci and Tetranacci-Lucas Quaternions and their Generating Functions, Binet's Formulas and Summations Formulas
In this section we define the bicomplex Tetranacci and Tetranacci-Lucas quaternions and give generating functions and Binet formulas for them. First, we give some information about bicomplex type quaternion sequences from the literature.
We now define bicomplex Tetranacci and Tetranacci-Lucas quaternions over the algebra BC.

Definition 1. The nth bicomplex Tetranacci quaternion is
and the nth Tetranacci-Lucas quaternion is It can be easily shown that {BCM n } n≥0 and {BCR n } n≥0 can also be defined by the recurrence relations: with the intial conditions The sequences {BCM n } n≥0 and {BCR n } n≥0 can be extended to negative subscripts by defining and for n = 1, 2, 3, . . ., respectively. Therefore, recurrences (2.3) and (2.4) hold for all integer n.
The first few bicomplex Tetranacci and Tetranacci-Lucas quaternions with positive subscript and negative subscript are given in Table 3 and Table 4: For two bicomplex Tetranacci quaternions BCM n and BCM k and for skaler λ ∈ R, the addition, substraction and multiplication with scalar are defined as componentwise, i.e., λBCM n = λM n + iλM n+1 + jλM n+2 + i jλM n+3 respectively, and product (multiplication) is defined as follows: Communications in Mathematics and Applications, Vol. 11, No. 1, pp. 95-112, 2020 Similarly, for two bicomplex Tetranacci-Lucas quaternions BCR n and BCR k and for skaler λ ∈ R, the addition, substraction and multiplication with scalar are defined as componentwise, i.e., respectively, and product (multiplication) is defined as follows: Moreover, three different conjugations for the bicomplex Tribonacci quaternion BCM n = M n + iM n+1 + jM n+2 + i jM n+3 are given as , and the squares of norms of the bicomplex Tribonacci quaternion are given by Similarly, we can give three different conjugations and the squares of norms for the bicomplex Tribonacci-Lucas quaternion BCR n = R n + iR n+1 + jR n+2 + i jR n+3 . Now, we will state Binet's formula for the bicomplex Tetranacci and Tetranacci-Lucas quaternions and in the rest of the paper we fix the following notations.
Note that using Binet's formula (1.5) of the Tetranacci numbers we have This proves (2.6). Similarly, we can obtain (2.5).
Next, we present generating functions. BCM n x n = (i + j + 2i j) respectively.
Proof. Let BCM n x n be generating function of the bicomplex Tetranacci quaternions. Then using the definition of the bicomplex Tetranacci quaternions, and substracting xg(x), x 2 g(x), x 3 g(x) and x 4 g(x) from g(x) and using the recurrence relation BCM n = BCM n−1 + BCM n−2 + BCM n−3 + BCM n−4 , we obtain

Now using
Similarly, we can obtain (2.9).
Next we present some summation formulas of Tetranacci numbers. In the following Lemma we present some summation formulas of Tetranacci-Lucas numbers.

Lemma 5.
For n ≥ 1 we have the following formulas: Next we present some summation formulas of bicomplex Tetranacci quaternions.

Theorem 6.
For n ≥ 0 we have the following formulas: Proof.
In the following Theorem, we give some summation formulas of bicomplex Tetranacci-Lucas quaternions.

Matrices and Determinants Related with Tetranacci and Tetranacci-Lucas Quaternions
We define the square matrix B of order 4 as: Induction proof may be used to establish Matrix formulation of M n and R n can be given as  Induction proofs may be used to establish the matrix formulations M n and R n . Now, we define the matrices B M and B R as These matrices B M and B R can be called bicomplex Tetranacci quaternion matrix and bicomplex Tetranacci-Lucas quaternion matrix, respectively.
Proof. We prove (a) by mathematical induction on n. If n = 0 then the result is clear. Now, we assume it is true for n = k, that is Thus, (3.4) holds for all non-negative integers n.

Corollary 9.
For n ≥ 0, the followings hold: Proof. The proof of (a) can be seen by the coefficient of the matrix B M and (3.1). The proof of (b) can be seen by the coefficient of the matrix B R and (3.1).

Five-Diagonal Matrix with Fourth Order Sequences and Applications
In this section we give another way to obtain nth term of the bicomplex Tetranacci and Tetranacci-Lucas quaternions. For this we need the following theorem.
Theorem 10. Let {x n } be any fourth-order linear sequence defined recursively as follows: x n = rx n−1 + sx n−2 + tx n−3 + ux n−4 , n ≥ 4 with the initial conditions x 0 = a, x 1 = b, x 2 = c, x 3 = d. Then for all n ≥ 0, we have  .
Proof. We proceed by induction on n. Since the equality holds for n = 0, 1, 2, 3. Now we assume that the equality is true for 4 ≤ k ≤ n. Then we will complete the inductive step n + 1 as follows: Note that  and and so    = rx n + sx n−1 + tx n−2 + u(−1) (n−1)+(n−1)  = rx n + sx n−1 + tx n−2 + ux n−3 .
This completes the inductive step and the proof of the theorem.
Note that in our cases r = s = t = u = 1. As a corallary of the above theorem, in the following we present another way to obtain nth term of the bicomplex Tetranacci and Tetranacci-Lucas quaternions.

Conclusion
Recently, there have been so many studies of the sequences of numbers in the literature and the sequences of numbers were widely used in many research areas, such as physics, engineering, architecture, nature and art. BC, the set of bicomplex numbers, is a complex Clifford Algebra which is the simplest example, and the only commutative one, see [13, p. 65]. If we use together sequences of integers defined recursively and bicomplex numbers, we obtain a new sequences such as bicomplex Fibonacci quaternions, bicomplex Pell quaternions, bicomplex Padovan quaternions.
Various families of bicomplex number (quaternion) sequences have been defined and studied by a number of authors. Some authors studied on second order bicomplex quaternion sequences such as [1], [2], [4], [8], [9] and some authors studied on third order bicomplex quaternion sequences such as [5], [12]. In this work, we introduce (fourth order) bicomplex Tetranacci and bicomplex Tetranacci-Lucas quaternions and prove some important properties of them. The methods used in this paper can be used for the other linear recurrence sequences, too. It is our intention to continue the study and explore properties of some type of bicomplex number sequences, such as bicomplex generalized Tetranacci quaternions and other fourth and fifth order bicomplex quaternions.