Unrestricted Tribonacci and Tribonacci–Lucas quaternions

: We define a generalization of Tribonacci and Tribonacci–Lucas quaternions with arbitrary Tribonacci numbers and Tribonacci–Lucas numbers coefficients, respectively. We get generating functions and Binet’s formulas for these quaternions. Furthermore, several sum formulas and a matrix representation are obtained.


Introduction
Quaternions have several applications in mathematics, see [13,29,37].The real quaternion algebra H has a basis {1, i, j, k} where Thus an element  ∈ H can be written as  =  + i + j + k.It is a division algebra which is not commutative.The real quaternion ℜ() − ℑ(), where ℜ() denotes the real part and ℑ() denotes the imaginary part of the quaternion, gives the conjugate of  and we denote this quaternion by  * .The multiplication of  and  * gives the norm of  and it is denoted by  ().Hence for  ̸ = 0, its inverse will be Many researchers are interested in quaternions with components chosen in number sequences.One of them is given in [18] and called as Fibonacci quaternions.The -th Fibonacci quaternion   is defined by where   is the -th Fibonacci number.The -th Lucas quaternion ℒ  is defined by where   is the -th Lucas number.After that, these structures attracted a lot of attention and various properties of such sequences were studied by many authors, see [4, 5, 11, 12, 14-17, 19-23, 25, 27, 28, 30-34, 36].
In [6], a generalized Tribonacci sequence {  } ≥0 is defined with initial conditions  0 = ,  1 = ,  2 =  and by the recurrence for , ,  integers and , ,  ∈ R. Then the well-known Tribonacci sequence denoted by {  }  is obtained for (, , ) = (1, 1, 1) and ( 0 ,  1 ,  2 ) = (1, 1, 2), see [9,10,24].On the other hand, the Tribonacci-Lucas sequence {  }  is obtained for (, , ) = (1, 1, 1) and ( 0 ,  1 ,  2 ) = (3, 1, 3), see [38].For  = −1+ √ 3 2 and the Binet formulae are see [38].Using the sequence {  } ≥0 , a quaternion sequence of order 3 is defined by, see [6].In [1], several properties of the classical Tribonacci quaternion sequence {   } ≥0 , which is defined by with Tribonacci number coefficients and the classical Tribonacci-Lucas quaternion sequence {︁  Q }︁ ≥0 , which is defined by with Tribonacci-Lucas number coefficients are presented.The Tribonacci quaternion sequence {   } ≥0 has the following generating function For  = 1 + i +  2 j +  3 k,  = 1 + i +  2 j +  3 k and  = 1 + i +  2 j +  3 k, the Binet formulae of the sequences {   } ≥0 and { Q } ≥0 can be obtained by using Equations ( 1) and (2) as respectively, see [1,6].In [26,35], some generalizations of Tribonacci quaternions are defined and several properties of these quaternions are presented.In [7], another generalization of the Tribonacci and Tribonacci-Lucas quaternion sequences is defined in the sense of polynomials.One of the common features of all these studies is that the coefficients of the studied quaternion sequences consist of consecutive terms of the selected number sequences.There are also some studies which takes the coefficients of quaternions from the number sequences randomly.In [8], a new class of quaternions whose coefficients are arbitrarily selected from the Fibonacci and Fibonacci-Lucas number sequences is defined.For arbitrary integers ,  and , the -th unrestricted Fibonacci and Lucas quaternions are defined by, respectively, [8].By definitions, choosing  =  =  = − we get the -th Fibonacci and -th Lucas numbers.For  = 1,  =  = − in (5), we obtain the -th Gaussian Fibonacci number.We also get the classical -th Fibonacci and Lucas quaternions from ( 5) and ( 6) as follows see [8].In [2], the coefficients of quaternions are chosen from Pell and Pell-Lucas numbers and in [3] the coefficients of quaternions are chosen from Fibonacci and Lucas hyper-complex numbers randomly.Inspiring from these studies, we will define a new class of quaternions whose coefficients are arbitrarily chosen from the Tribonacci and Tribonacci-Lucas number sequences in this paper.We also investigate several properties and identities of defined sequences.

Unrestricted Tribonacci and Tribonacci-Lucas quaternions
In this section, a generalization of the classical Tribonacci quaternions and Tribonacci-Lucas quaternions are defined and their Binet's formulas and generating functions are obtained.Let ,  and  be arbitrary positive integers throughout the paper.
For  ≥ 0, we have By definitions, we get the well-known sequences by taking special values for ,  and  such as: where    and  Q are defined in (3) and (4).We now present the Binet formulas and generating functions of   3  .

Some identities of unrestricted Tribonacci and Tribonacci-Lucas quaternions
In this part, we will present some identities of  (,,)  and Q(,,)

𝑛
. Since , ,  are arbitrary positive integers we obtain some identities given in [1] for  = 1,  = 2,  = 3.We give the proofs of the first identities given in theorems and the other ones can be obtained similarly.( (,,) Proof.Using the definition, we can obtain (7) as follows Hence we get the result.Theorem 3.2.

𝒬 (𝑝,𝑟,𝑠) 𝑚+𝑛
=  (,,) where Proof.Since the Tribonacci numbers and Tribonacci-Lucas numbers satisfy the following equalities, see [38], we have  (,,) The following theorem gives some finite sum identities of defined quaternions.From the hypothesis, we can write Hence, the results follows by induction. Let with initial conditions  0 =  1 = 0. Using the definitions of these sequences, we define the related quaternion sequences as follows:

A matrix representation
For  (,,) , we present a matrix generator as follows.