Hyperbolic -Oresme and -Oresme-Lucas Quaternions
ENGIN OZKAN1, HAKAN AKKUS2
1Department of Mathematics
Faculty of Sciences
Marmara University
İstanbul
TURKIYE
²Department of Mathematics,
Graduate School of Natural and Applied Sciences,
Erzincan Binali Yıldırım University,
Erzincan,
TURKIYE
Abstract: - In this study, we define hyperbolic -Oresme and -Oresme-Lucas quaternions. For these
quaternions, we give the Binet Formulas, summation formulas, etc. Then we obtain the generating functions
and exponential generating functions of these quaternions. Also, we find relations among the hyperbolic -
Oresme and -Oresme-Lucas quaternions and their conjugates. In addition, we calculate the special identities
of these quaternions. Moreover, we examine the relationships between the hyperbolic -Oresme and -Oresme-
Lucas quaternions. Finally, the terms of the -Oresme and -Oresme-Lucas sequences are associated with their
hyperbolic quaternion values.
Key-Words: - Oresme number, Quaternions, Lucas Number, Catalan identity, Generating function
Received: May 13, 2024. Revised: October 14, 2024. Accepted: November 17, 2024. Published: December 27, 2024.
1 Introduction
Sequences of numbers play a vital role in
understanding the complexity of any problem
consisting of some patterns. An example of this is
the Rabbit problem in Leonardo Fibonacci's classic
book Liber Abaci. Inspired by the rabbit problem,
the Fibonacci sequence was developed and the
relationship between the terms of this sequence
became the golden ratio. The Fibonacci and Lucas
sequences are famous sequences of numbers. These
sequences have intrigued scientists for a long time.
Fibonacci and Lucas sequences have been applied in
various fields such as Algebraic Coding Theory [1],
Phyllotaxis [2], Biomathematics [3], Computer
Science [4], etc. Many generalizations of the
Fibonacci sequence have been given. The known
examples of such sequences are the Bronze
Fibonacci, Bronze Lucas, -Fibonacci, -Lucas,
Oresme, -Chebsyhev, Jacobsthal-Lucas, Pell,
Lenardo, Narayana, Padovan sequences, etc (see for
details in [5], [6], [7], [8], [9], [10], [11], [12], [13],
[14], [15], [16], [17]).
For , the Fibonacci numbers and Lucas
numbers are defined by the recurrence relations,
respectively,
, and ,
with the initial conditions , and
, .
Binet formulas Fibonacci numbers and Lucas
numbers are given by relations, respectively,
and ,
where
and
are the roots of the
characteristic equation . Here the
number is the known golden ratio.
In [18], for , he defined the Oresme sequence
and Oresme-Lucas sequence by the
recurrence relations, respectively;
, and
,
with the initial conditions,
and
, .
With the help of the recurrence relation of the
Fibonacci sequence, -sequences were introduced
and these sequences had an important place in
number theory [19].
PROOF
DOI: 10.37394/232020.2024.4.14
Engin Ozkan, Hakan Akkus
E-ISSN: 2732-9941
141
Volume 4, 2024
In [20], for , they defined the -Oresme
sequence and -Oresme-Lucas sequence
the recurrence relations, respectively;
, and
,
with the initial conditions ,
and
, .
Binet formulas of the -Oresme and the-Oresme-
Lucas sequence are given by relations, respectively,
=
and ,
where
and
are the roots
of the characteristic equation
.
In mathematics, quaternions (or quadruplets) are a
number system that expands the complex numbers
into one real and three imaginary dimensions.
The quaternions were first described by Hamilton in
1843. In addition, quaternions are used to control
rotational movements, especially in Kinematics
[21], 3D games [22], mechanics [23], Eulerian
angles [24], and Chemistry [25]. In [26], Horadam
defined Complex Fibonacci and Fibonacci
quaternions, and various features were found.
The algebra of hyperbolic quaternions is an algebra
that is not related to the elements of the form over
the real numbers.
,
He gave the properties of the components defined
in Table 1.
Table 1. Hyperbolic Quaternions Units
In [27], he did a lot of research on hyperbolic
quaternions and their properties. An expression of
the general form of hyperbolic quaternions is
.
Here, are the terms of the sequence
and are hyperbolic quaternions.
In [28,] he defined the hyperbolic -Fibonacci and
-Lucas quaternions and he found properties of
these quaternions. Also, they conducted a study on
the quaternions and obtained many features related
to these quaternions [29]. In addition, they
introduced the Jacobsthal and Jacobsthal-Lucas
quaternions [30]. Moreover, many applications of
sequences have been made on quaternions ( see for
details in [31], [32], [33], [34], [35], [36], [37], [38],
[39], [40]).
As seen above, many generalizations of hyperbolic
quaternions of sequences have been given so far. In
this study, we give new generalizations inspired by
the hyperbolic -Fibonacci and Jacobsthal
quaternions. We call these quaternions the
hyperbolic -Oresme and -Oresme quaternions
and denote them as
, and
, respectively.
We separate the article into three parts.
In Chapter 2, we define the hyperbolic -Oresme
and -Oresme-Lucas quaternions and the terms of
these quaternions are given. Then, we find some
properties of these quaternions.
In Chapter 3, information is given about the
characteristic equations of hyperbolic -Oresme and
-Oresme-Lucas quaternions. Then, we obtain the
Binet formulas, generating functions and summation
formulas of these quaternions. In addition, we find
the relationship of hyperbolic -Oresme and -
Oresme-Lucas quaternions, Catalan identity, Cassini
identity, D’ocagne identity, Vajda’s identity, etc.
Finally, the terms of the sequence are associated
with their hyperbolic values.
2 Hyperbolic -Oresme and -
Oresme-Lucas Quaternions
For , the hyperbolic -Oresme
and -
Oresme-Lucas quaternions
are defined by,
respectively,
and
where , is -Oresme sequence, , -
Oresme-Lucas sequence and , and are the
hyperbolic quaternion units in table 1.
Let us now give the first three terms of the
hyperbolic -Oresme and -Oresme-Lucas
quaternions, respectively,
,
,
,
and
,
,
.
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DOI: 10.37394/232020.2024.4.14
Engin Ozkan, Hakan Akkus
E-ISSN: 2732-9941
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.
Definition 2.1. For , the conjugate of
hyperbolic -Oresme
and -Oresme-Lucas
quaternions are defined by, respectively,
and
.
Definition 2.2. For , the norms of the
hyperbolic -Oresme
and -Oresme-Lucas
quaternions
are defined by, respectively,
and
.
In the following theorems, we examine the relations
between the
,
,
, and
quaternions.
Theorem 2.1. Let . The following equations
are true:
i.
,
ii.
,
iii.
,
iv.
.
Proof. i. If the definition is used, we have
.
Since,
. Thus, we obtain
.
The proofs of the others are shown similarly.
Theorem 2.2. We obtain
i.
,
ii.
,
iii.
,
iv.
v.
vi.
.
Proof. ii. If the definition is used, we have
.
Thus, we obtain
.
The proofs of the others are shown similarly.
Theorem 2.3. We obtain
i.
.
ii.
.
Proof. i. If the definition is used, we have
.
The proofs of the others are shown similarly.
3 Properties of Hyperbolic -Oresme
and -Oresme-Lucas Quaternions
In this chapter, we obtain properties of the
hyperbolic -Oresme and -Oresme-Lucas
quaternions. Then, we examine the relationships
between these quaternions. Also, we calculate the
special identities of these quaternions. In addition,
we find the terms of the -Oresme and -Oresme-
Lucas sequences are associated with their
hyperbolic quaternion values.
In the following theorem, the Binet formulas of the
,
,
, and
quaternions are
expressed.
Theorem 3.1. Let . We obtain
i.
, ii.
,
iii.
, iv.
,
where
,
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DOI: 10.37394/232020.2024.4.14
Engin Ozkan, Hakan Akkus
E-ISSN: 2732-9941
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,
and
.
Proof. i. With the help of the characteristic
equation, the following results are obtained.
,,
,
and
.
The Binet form of the hyperbolic -Oresme
quaternions is
.
With the initial conditions, the following equations
are obtained.
and
.
Thus, we obtain
.
So, we have
.
The proofs of the others are shown similarly.
In the following theorems, we give special sum
formulas of the
, and
quaternions.
Theorem 3.2. Let and . We obtain
i.
ii.
.
Proof. i. Using the definition, we have
.
Since
, we
get
.
So, we obtain
.
The proof of the other is shown similarly.
Theorem 3.3. Let and . We obtain
i.
ii.
,
iii.
,
iv.
.
Proof. With the help of definitions, Binet formulas,
and geometric series, we have
i.
Thus, we get
.
The proofs of the others are shown similarly.
In the following theorems, we give special
generating functions of the
, and
quaternions.
Theorem 3.4. The generating functions for
hyperbolic -Oresme and -Oresme-Lucas
sequences are given as follows, respectively,
and
.
Proof. The following equations are written for the
hyperbolic -Oresme sequence.
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DOI: 10.37394/232020.2024.4.14
Engin Ozkan, Hakan Akkus
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.
Thus, we obtain
.
Similarly to i, is obtained.
Theorem 3.5. For , and , we
obtain
i.
,
ii.
,
iii.
,
iv.
,
v.
,
vi.
.
Proof. With the help of definitions, Binet formulas,
and geometric series, we have
i.
.
Thus, we get
.
The proof of the other is shown similarly.
In the next lemma, we give the properties that will
be used to prove many theorems.
Lemma 3.1. We have
i.
,
ii.
,
iii.
,
iv.
,
v.
,
vi.
,
vii.
,
viii.
,
ix. ,
x. ,
xi.
,
xii. .
Proof. v. If the definition is used, we have
.
Thus,
.
Other proofs are shown using definitions.
In the following theorems, we calculate some
identities for
, and
quaternions.
Theorem 3.6. (Cassini Identity) Let . We get
i.
,
ii.
.
Proof. i. With the Binet formula, we get
)(
.
Then, we have
.
So, we obtain
.
ii. With the Binet formula, we obtain
.
Then, we get
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DOI: 10.37394/232020.2024.4.14
Engin Ozkan, Hakan Akkus
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Volume 4, 2024
.
Thus, we have
.
Theorem 3.7. (Catalan Identity) Let . We
get
i.
),
ii.
).
Theorem 3.8. (Vajda’s Identity) Let .
We obtain
i.
),
ii.
.
Theorem 3.9. (d’Ocagne Identity) Let .
. We obtain
i.
),
ii.
.
The proofs of Theorem 3.7.,-3.9., are similar to
Theorem 3.6., using Binet formulas, Lemma 3.1.,
and definitions.
In the following theorems, we examine the
relationships between hyperbolic -Oresme and -
Oresme-Lucas quaternions.
Theorem 3.10. Let . The following equations
are true:
i.
,
ii.
,
iii.
,
iv.
.
Proof. i. The following relation is used for proofs;
.
For these values, we obtain;
,
.
We find
and
.
Thus, we obtain
.
The proofs of the others are shown similarly.
Theorem 3.11. For any integer , we get
i.
,
ii.
,
iii.
.
Proof. With the Binet formula, we obtain
i.
.
Thus, we get
.
The proofs of the others are shown similarly.
In the following theorems, we associate the terms of
the -Oresme and -Oresme-Lucas sequences with
their hyperbolic quaternion values.
Theorem 3.12. Let . . We
obtain
i.
,
ii.
,
iii.
.
Proof. i. If Binet formulas are used, we get
.
Thus, we have
.
The proofs of the others are shown similarly in i.
Theorem 3.13. For all we have
i.
,
ii.
,
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Engin Ozkan, Hakan Akkus
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iii.
,
iv.
.
Proof. i. If Binet formulas are used, we get
.
Since, and
. Thus, we obtain
.
The proofs of the others are shown similarly in i.
Theorem 3.14. For all , we obtain
i.
,
ii.
,
iii.
,
iv.
.
Proof. i. If Binet formulas are used, we get
.
Since, and
. Thus, we obtain
.
The proofs of the others are shown similarly in i.
Theorem 3.15. If and , we obtain
i.
,
ii.
,
iii.
,
iv.
.
Proof. iv. With the Binet formula, we have
Since,
. Thus, we get
.
Theorem 3.16. Let and . We
obtain
i.
,
ii.
.
Proof. With the Binet formulas, we have
i.
.
Thus, we obtain
.
ii.
Thus, we get
.
4 Conclusion
In this study, we defined the hyperbolic -Oresme
and -Oresme-Lucas quaternions. Then, we
obtained some properties of these quaternions. Also,
we examined the relationships between these
quaternions. In addition, we found relations among
the hyperbolic -Oresme and -Oresme-Lucas
quaternions and their conjugates. Furthermore, we
calculated the special identities of these quaternions.
Moreover, we found the terms of the -Oresme and
-Oresme-Lucas sequences are associated with their
hyperbolic quaternion values. In the future, we can
spread a new approach to hyperbolic -Oresme and
-Oresme-Lucas octonions and sedenions.
Acknowledgments
The authors would like to thank the referees who
carefully read the article and made valuable
comments.
PROOF
DOI: 10.37394/232020.2024.4.14
Engin Ozkan, Hakan Akkus
E-ISSN: 2732-9941
147
Volume 4, 2024
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
-ENGIN OZKAN carried out the introduction and
the main result of the article.
-Hakan AKKUS has improved Chapter 2 and
Chapter 3.
-All authors read and approved the final manuscript.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This study did not receive any funding in any form.
Conflict of Interest
The authors have no conflict of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
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DOI: 10.37394/232020.2024.4.14
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