Tetranacci and Tetranacci-Lucas Quaternions
Y\"uksel Soykan

TL;DR
This paper introduces Tetranacci and Tetranacci-Lucas quaternions, exploring their properties and relationships within the algebraic structure of quaternions, expanding the mathematical understanding of these special sequences.
Contribution
The paper presents the first definition and analysis of Tetranacci and Tetranacci-Lucas quaternions, including their properties and interrelations.
Findings
Defined Tetranacci and Tetranacci-Lucas quaternions
Derived properties and identities of these quaternions
Established relationships between Tetranacci and Tetranacci-Lucas quaternions
Abstract
The quaternions form a 4-dimensional Cayley-Dickson algebra. In this paper, we introduce the Tetranacci and Tetranacci-Lucas quaternions. Furthermore, we present some properties of these quaternions and derive relationships between them.
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Taxonomy
TopicsAdvanced Mathematical Theories and Applications · Algebraic and Geometric Analysis · Mathematics and Applications
Tetranacci and
Tetranacci-Lucas Quaternions
Yüksel Soykan
Department of Mathematics, Art and Science Faculty,
Zonguldak Bülent Ecevit University, 67100, Zonguldak, Turkey
Abstract. The quaternions form a -dimensional Cayley-Dickson algebra. In this paper, we introduce the Tetranacci and Tetranacci-Lucas quaternions. Furthermore, we present some properties of these quaternions and derive relationships between them.
2010 Mathematics Subject Classification. 11B39, 11B83, 17A45, 05A15.
**Keywords. **Tetranacci numbers, quaternions, Tetranacci quaternions, Tetranacci-Lucas quaternions.
1. Introduction
Tetranacci sequence and Tetranacci-Lucas sequence are defined by the fourth-order recurrence relations
[TABLE]
and
[TABLE]
respectively. is the sequence in [References] and is the sequence in [References]. This sequence has been studied by many authors and more detail can be found in the extensive literature dedicated to these sequences, see for example [References], [References], [References], [References], [References], [References].
The sequences and can be extended to negative subscripts by defining
[TABLE]
and
[TABLE]
for respectively. Therefore, recurrences (1.1) and (1.2) hold for all integer
We can write (1.1) as Substracting this from (1.1), we see that Tetranacci numbers also satisfy the following useful alternative linear recurrence relation for :
[TABLE]
Extension of the definition of to negative subscripts can be proved by writing the recurrence relation (1.3) as
[TABLE]
Similarly, we have
[TABLE]
The following Table 1 presents the first few values of the Tetranacci and Tetranacci-Lucas numbers with positive and negative subscripts:
Table 1. Tetranacci and Tetranacci-Lucas Numbers with non-negative and negative indices
[TABLE]
It is well known that for all integers usual Tetranacci and Tetranacci-Lucas numbers can be expressed using Binet’s formulas
[TABLE]
(see for example [References] or [References])
or
[TABLE]
(see for example [References]) and
[TABLE]
respectively, where and are the roots of the cubic equation Moreover,
[TABLE]
where
[TABLE]
Note that we have the following identities:
[TABLE]
Note that the Binet form of a sequence satisfying (1.1) and (1.2) for non-negative integers is valid for all integers This result of Howard and Saidak [References] is even true in the case of higher-order recurrence relations as the following theorem shows.
Theorem 1** ([References]).**
Let be a sequence such that
[TABLE]
for all integers with arbitrary initial conditions Assume that each and the initial conditions are complex numbers. Write
[TABLE]
with and distinct. Then
**(a): **
*For all *
[TABLE]
where
[TABLE]
with each a constant determined by the initial conditions for . Here, equation (1.9) is called the Binet form (or Binet formula) for We assume that so that can be extended to negative integers
If the zeros of (1) are distinct, as they are in our examples, then
[TABLE]
**(b): **
The Binet form for is valid for all integers
The generating functions for the Tetranacci sequence and Tetranacci-Lucas sequence are
[TABLE]
respectively.
In this paper, we define Tetranacci and Tetranacci-Lucas quaternions in the next section and give some properties of them. Before giving their definition, we present some information on quaternions.
Quaternions were invented by Irish mathematician W. R. Hamilton (1805-1865) as an extension to the complex numbers. Most mathematicians have heard the story of how Hamilton invented the quaternions. The 16th of October 1843 was a momentous day in the history of mathematics and in particular a major turning point in the subject of algebra. On that day William Rowan Hamilton had a brain wave and came up with the idea of the quaternions. He carved the multiplication formulae with his knife into the stone of the Brougham Bridge (nowadays known as Broomebridge) in Dublin,
[TABLE]
One reason this story is so well-known is that Hamilton spent the rest of his life obsessed with the quaternions and their applications to geometry. The story of this discovery has been translated into many different languages. For this story and for a full biography of Hamilton, we refer the work of Hankins [References].
After the middle of the 20th century, the practical use of quaternions has been discovered in comparison with other methods and there has been an increasing interest in algebra problems on quaternion field since many algebra problems on quaternion field were encountered in some applied and pure science such as the quantum physics, computer science, analysis and differential geometry.
A quaternion is a hyper-complex number and is defined by
[TABLE]
where a_{0},a_{1},a_{2}\and are real numbers or scalers and are the standard orthonormal basis in . The set of all quaternions are denoted by . Note that we can write
[TABLE]
where . and are called the scalar part and the vector part of the quaternion respectively. The are called the components of the quaternion .
Addition of quaternions is defined as componentwise and the quaternion multiplication is defined as follows:
[TABLE]
Note that from (1.10), we have
[TABLE]
So, multiplication on is not commutative. The identities in (1.10) and (1.11), sometimes are known as Hamilton’s rules. Quaternions have the following multiplication Table 2:
Table 2. Multiplication Table
The product of two quaternions and is
[TABLE]
The conjugate of the quaternion is defined by
[TABLE]
For two quaternions we have
[TABLE]
The norm of a quaternion is defined by
[TABLE]
The norm is multiplicative:
[TABLE]
Division is uniquely defined (except by zero), thus quaternions form a division algebra. For two quaternions we have
[TABLE]
The inverse (reciprocal) of a nonzero quaternion is given by
[TABLE]
In A. Hurwitz proved that the only real composition algebras are , , and (here stands for octonion algebras). (A real composition algebra is an algebra over , not necessarily associative or finite-dimensional, equipped with a nonsingular quadratic form such that for all . The form is given by the norm. For more information on quadratic form, see [References, pp. 44 and 53])
Briefly , the algebra of quaternions, has the following properties:
- •
is a dimensional non-commutative (Carley-Dickson) algebra over the reals.
- •
is an associative algebra.
- •
is a division algebra, i.e. an algebra which is also a division ring, i.e., each nonzero element of is invertible.
- •
is a composition algebra.
- •
is a flexible algebra, i.e. for all .
- •
is an alternative algebra, i.e. they have the property and for all .
For the basics on the quaternions theory, we refer the work of Ward [References] and Lewis [References].
We remark that
- •
, , and are the only normed division algebras.
- •
, , and are the only alternative division algebras.
Last two properties shows what is so great about , , and . For this two properties and their histories, see [References].
2. The Tetranacci and Tetranacci-Lucas Quaternions and their
Generating Functions, Binet’s Formulas and Summations Formulas
In this section, we define Tetranacci and Tetranacci-Lucas quaternions and give generating functions and Binet formulas for them. First, we give some information about quaternion sequences from the literature.
There are various types of quaternion sequences which have been studied by many researchers. Horadam [References] introduced th Fibonacci and th Lucas quaternions as
[TABLE]
and
[TABLE]
respectively, where and are the th Fibonacci and Lucas numbers respectively. He also defined generalized Fibonacci quaternion as
[TABLE]
where is the th generalized Fibonacci number (which is now called Horadam number) by the recursive relation ( and are arbitrary integers). Halici [References] gave the generating functions and Binet formulas for the Fibonacci and Lucas quaternions.
Cerda-Morales [References] defined and studied the generalized Tribonacci quaternion sequence that includes the previously introduced Tribonacci, Padovan, Narayana and third order Jacobsthal quaternion sequences. In [References], the author defined generalized Tribonacci quaternion as
[TABLE]
where is the th generalized Tribonacci number defined by the third-order recurrance relations
[TABLE]
here are arbitrary integers and are real numbers.
Many other generalizations of Fibonacci quaternions have been given, see for example Catarino [References], Halici and Karataş [References], and Polatlı [References], Szynal-Liana and Wloch [References] and Tasci [References] for second order quaternion sequences and Akkus and Kızılaslan [References], Szynal-Liana and Wloch [References], Tasci [References], Cerda-Morales [References] for third order quaternion sequences.
We now define Tetranacci and Tetranacci-Lucas quaternions over the quaternion algebra . The th Tetranacci quaternion is
[TABLE]
and the th Tetranacci-Lucas quaternion is
[TABLE]
It can be easily shown that
[TABLE]
and
[TABLE]
Note that
[TABLE]
and
[TABLE]
The conjugate of and are defined by
[TABLE]
and
[TABLE]
respectively.
Now, we will state Binet’s formula for the Tetranacci and Tetranacci-Lucas quaternions and in the rest of the paper we fix the following notations.
[TABLE]
Theorem 2**.**
(Binet’s Formulas) For any integer the th Tetranacci quaternion is
[TABLE]
and the th Tetranacci-Lucas quaternion is
[TABLE]
Proof. Using Binet’s formula of the Tetranacci-Lucas numbers, we have
[TABLE]
Note that using Binet’s formula (1.7) of the Tetranacci numbers we have
[TABLE]
This proves (2.6). Similarly, we can obtain (2).
Remark 3**.**
According to Theorem 1, Binet’s Formulas of the Tetranacci and Tetranacci-Lucas quaternions are true for all integers
Next, we present generating functions.
Theorem 4**.**
The generating functions for the Tetranacci and Tetranacci-Lucas quaternions are
[TABLE]
and
[TABLE]
respectively.
Proof. Let
[TABLE]
be generating function of the Tetranacci quaternions. Then, using the definition of the Tetranacci quaternions, and substracting and from we obtain (note the shift in the index in the third line)
[TABLE]
Note that we used the recurrence relation . Rearranging above equation, we get
[TABLE]
or
[TABLE]
since . Now using
[TABLE]
we obtain
[TABLE]
Similarly, we can obtain (2.9).
In the following theorem, we present another forms of Binet’s formulas for the Tetranacci and Tetranacci-Lucas quaternions using generating functions.
Theorem 5**.**
For any integer the th Tetranacci quaternion is
[TABLE]
and the th Tetranacci-Lucas quaternion is
[TABLE]
Proof. We can use generating functions. Since the roots of the equation are and
[TABLE]
we can write the generating function of as
[TABLE]
We need to find and , so the following system of equations should be solved:
[TABLE]
Then, we find that
[TABLE]
and
[TABLE]
Thus, from this, we obtain Binet’s formula of Tetranacci quaternion. Similarly, we can obtain Binet’s formula of the Tetranacci-Lucas quaternion.
If we compare Theorem 2 and Theorem 5 and use the definition of we have the following Remark showing relations between and
Remark 6**.**
We have the following identities:
**(a): **
[TABLE]
**(b): **
[TABLE]
Now, we present the formulas which give the summation of the first Tetranacci and Tetranacci-Lucas numbers.
Lemma 7**.**
For every integer we have
[TABLE]
and
[TABLE]
Proof. (2.11) and (2.12) are given in Soykan [References, Corollaries 2.7 and 2.8].
Note that (2.11) and (2.12) can be easily proved by mathematical induction as well.
Next, we present the formulas which give the summation of the first Tetranacci and Tetranacci-Lucas quaternions.
Theorem 8**.**
The summation formula for Tetranacci and Tetranacci-Lucas quaternions are
[TABLE]
and
[TABLE]
Proof. Using (2.1) and (2.11), we obtain
[TABLE]
and so
[TABLE]
where
[TABLE]
Hence
[TABLE]
This proves (2.13). Similarly, we can obtain (2.14).
Note that above Theorem can be proved by induction as well.
Theorem 9**.**
For we have the following formulas:
**(a): **
**
**(b): **
**
*Proof. *The proof follows from the following identities:
[TABLE]
and
[TABLE]
(2.15) and (2.16) are given in Soykan [References, Corollary 2.7].
Note that (2.15) and (2.16) can be easily proved by mathematical induction as well. Of course, the above theorem itself can be proved by induction.
Theorem 10**.**
For we have the following formulas:
**(a): **
**
**(b): **
**
*Proof. *The proof follows from the following identities:
[TABLE]
and
[TABLE]
(2.17) and (2.18) are given in Soykan [References, Corollary 2.8].
Note that (2.17) and (2.18) can be easily proved by mathematical induction as well. Of course, the above theorem itself can be proved by induction.
3. Matrices and Determinants related with Tetranacci and
Tetranacci-Lucas Quaternions
Define the determinants and for all integers by
[TABLE]
Theorem 11**.**
The following statements are true.
**(a): **
* and for all integers *
**(b): **
**
**(c): **
**
Proof. (a) is a special case of a result in [References]. Expanding along the top row gives 563M_{n}=86R_{n+3}-61R_{n+2}-71R_{n+1}-87R_{n}\and now (b) follows. Expanding along the top row gives R_{n}=6M_{n+1}-M_{n}-M_{n+3}\and now (c) follows.
Consider the sequence which is defined by the fourth-order recurrence relation
[TABLE]
The numbers can be expressed using Binet’s formula
[TABLE]
We define the square matrix of order as:
[TABLE]
such that
Induction proof may be used to establish
[TABLE]
Matrix formulation of and can be given as
[TABLE]
and
[TABLE]
Induction proofs may be used to establish the matrix formulations and .
Now we define the matrices and as
[TABLE]
These matrices and can be called Tetranacci quaternion matrix and Tetranacci-Lucas quaternion matrix, respectively.
Theorem 12**.**
For the followings are valid:
**(a): **
[TABLE]
**(b): **
[TABLE]
Proof. We prove (a) by mathematical induction on If then the result is clear. Now, we assume it is true for that is
[TABLE]
If we use (2.3), then we have Then, by induction hypothesis, we obtain
[TABLE]
Thus, (3.4) holds for all non-negative integers
(3.5) can be similarly proved .
Corollary 13**.**
For the followings hold:
**(a): **
**
**(b): **
**
Proof. The proof of (a) can be seen by the coefficient of the matrix and (3.1). The proof of (b) can be seen by the coefficient of the matrix and (3.1).
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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- 2[2] Akkus, I., and Kızılaslan, G., On Some Properties of Tribonacci Quaternions, ar Xiv:1708.05367 v 1 [math.CO] 17 Aug 2017.
- 3[3] Catarino, P., The Modified Pell and Modified k-Pell Quaternions and Octonions. Advances in Applied Clifford Algebras 26, 577-590, 2016.
- 4[4] Cerda-Morales, G., On a Generalization for Tribonacci Quaternions, Mediterranean Journal of Mathematics, 14 (239), 1–12, 2017.
- 5[5] Cerda-Morales, G., Identities for Third Order Jacobsthal Quaternions, Adv. Appl. Clifford Algebras 27, 1043-1053, 2017.
- 6[6] Dresden, G. P., Du, Z., A Simplified Binet Formula for k-Generalized Fibonacci Numbers, J. Integer Seq. 17, art. 14.4.7, 9 pp., 2014.
- 7[7] Halici, S., On Fibonacci Quaternions, Adv. Appl. Cliff ord Algebras 22, 321–327, 2012.
- 8[8] Halici, S., Karataş, A., On a Generalization for Fibonacci Quaternions. Chaos Solitons and Fractals 98, 178–182, 2017.
