Hyperbolic -Fibonacci and -Lucas Quaternions
HAKAN AKKUS1, ENGIN OZKAN2
1Department of Mathematics,
Graduate School of Natural and Applied Sciences,
Erzincan Binali Yıldırım University,
Erzincan
TÜRKIYE
²Department of Mathematics,
Faculty of Sciences,
Marmara University,
İstanbul
TÜRKIYE
Abstract: - In this study, we define hyperbolic -Fibonacci and -Lucas quaternions. For these
hyperbolic quaternions, we give the special summation formulas, special generating functions, etc. Also, we
calculate the special identities of these hyperbolic quaternions. In addition, we obtain the Binet formulas in two
different ways. The first is in the known classical way and the second is with the help of the sequence's
generating functions. Moreover, we examine the relationships between the hyperbolic -Fibonacci and
-Lucas quaternions. Finally, the terms of the -Fibonacci and -Lucas sequences are associated
with their hyperbolic quaternion values.
Key-Words: - sequence, Fibonacci numbers, Hyperbolic Quaternions, Generating Function, Cassini
Identity, Summation formula
1 Introduction
The Fibonacci, Lucas, and Pell sequences are
famous sequences of numbers. These sequences
have intrigued scientists for a long time. Fibonacci
sequences have been applied in various fields such
as Algebraic Coding Theory [1], [2], graph theory
[3], [4], Biomathematics [5], Computer Science [6],
and so on. Many generalizations of the Fibonacci
sequence have been given. The known examples of
such sequences are the Pell, Pell-Lucas, -
Fibonacci, -Jacobsthal-Lucas, -Lucas, -Pell, -
Pell-Lucas, and Modified -Pell sequences, etc (see
for details in [7], [8], [9], [10], [11]).
For , the Fibonacci numbers , Lucas
numbers , Pell numbers , and Jacobsthal-Lucas
numbers are defined by the recurrence relations,
respectively,
, ,
, and
with the initial conditions , ,,
, , , and, .
For , , , and the Binet formulas are given
by relations, respectively,
, ,
, and
where
,
, , and
are the roots of the characteristic equation
and , respectively.
Here and numbers are the known golden ratio
and silver ratio.
With the help of the recurrence relation of the
Fibonacci sequence, -sequences were
introduced, and these sequences had an important
place in number theory.
In [12], [13], for they defined the -
Fibonacci and -Lucas sequences by the
recurrence relations, respectively,
Received: April 24, 2024. Revised: September 15, 2024. Accepted: October 16, 2024. Published: November 14, 2024.
PROOF
DOI: 10.37394/232020.2024.4.9
Hakan Akkus, Engin Ozkan
E-ISSN: 2732-9941
97
Volume 4, 2024
,
with the initial conditions , and
, . In addition, they found the Binet
formulas and properties of these sequences. In [14],
they did applied work on the matrix representations
of Fibonacci sequences. Also, in [15], she
defined the -Jacobsthal and -Jacobsthal-
Lucas sequences and she found the generating
function, the Binet formulas and some features of
these sequences.
The quaternions were first described by Hamilton in
1843. Then, quaternions used to control rotational
movements especially in 3D games and Eulerian
angles. In [16], he defined Complex Fibonacci and
Fibonacci quaternions and he were found various
features of these sequences.
The algebra of hyperbolic quaternions is an algebra
that is not related to the elements of the form over
the real numbers.
,
In [16], he gave the properties of the units defined
as in Table 1.
Table 1. Hyperbolic Quaternions Units
In [17], he did a lot of research on hyperbolic
quaternions and their properties. An expression of
the general form of hyperbolic quaternions is
.
Here, the terms of the sequence,
are hyperbolic quaternions.
In [18], he defined the hyperbolic -Fibonacci and
-Lucas quaternions and he found properties of
these quaternions. Also, they conducted a study on
the hyperbolic Leonardo and Francois quaternions
and obtained many features related to these
quaternions [19]. In addition, they introduced the
Jacobsthal and Jacobsthal-Lucas quaternions [20].
Moreover, they did many studies on hyperbolic
quaternions,octonions, and sedenions [21], [22],
[23], [24], [25] [26], [27] [28], [29], [30].
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 quaternions and
Jacobsthal and Jacobsthal-Lucas quaternions. We
call these quaternions the hyperbolic -
Fibonacci and -Lucas quaternions and denote
them as
, and
, respectively.
We separate the article into three parts.
In chapter 2, we define the hyperbolic -Pell and
-Pell-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 -
Fibonacci and -Lucas quaternions. Then, we
obtain the Binet formulas, generating functions, and
sum of terms of these quaternions. In addition, we
examine the relationship of hyperbolic -
Fibonacci and -Lucas quaternions. Moreover,
we calculate the special identities of these
quaternions. Finally, we associate the terms of the
-Fibonacci and -Lucas sequences with
their hyperbolic quaternion values.
2 Hyperbolic -Fibonacci and
-Lucas Quaternions
For , the hyperbolic -Fibonacci
and -Lucas quaternions
are defined
by, respectively,
and
where is -Fibonacci sequence,
-Lucas sequence and and are the
hyperbolic quaternion units in table 1.
Let us now give some terms of the hyperbolic -
Fibonacci and -Lucas quaternions below.
,
,
,
and
,
,
.
Definition 2.1. For , the conjugate of -
Fibonacci
and -Lucas
quaternions are defined by, respectively,
and
.
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DOI: 10.37394/232020.2024.4.9
Hakan Akkus, Engin Ozkan
E-ISSN: 2732-9941
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Volume 4, 2024
.
Definition 2.2. For , the norms of the
hyperbolic -Fibonacci
and -
Lucas
quaternions are defined by,
respectively,
and
.
3 Properties of Hyperbolic -
Fibonacci and -Lucas
Quaternions
In this chapter, the relationships between the
hyperbolic -Fibonacci and -Lucas
quaternions are examined. In addition, some
identities are obtained.
Theorem 3.1. For , the hyperbolic -
Fibonacci and -Lucas quaternions provide the
following recurrence relations.
i.
,
ii.
,
iii.
,
iv.
,
v.
,
vi.
.
Proof. If the definition of the hyperbolic function is
used, we have
i.
.
Since, .
Thus, we obtain
.
The proofs of the others can be given in the same
way.
In the following theorem, the Binet formulas of the
,
,
, and
quaternions are
expressed.
Theorem 3.2. Let . We obtain
i.
, ii.
,
iii.
, iv.
where
,
,
,
.
Proof. i. With the help of the characteristic
equation, the following results are obtained.
,,
and
.
The Binet form of the hyperbolic -Fibonacci
quaternions is
.
With the initial conditions, the following equations
are obtained.
and
.
Thus, we obtain
and
.
So, we have
The proofs of the others can be given in the same
way.
In the next Lemma, we obtain the properties that
will be used in the proof of many theorems.
Lemma 3.1. We have
i.
,
ii.
,
iii.
,
iv.
v.
,
vi.
,
vii. ,
viii. ,
ix.
,
x.
.
Proof. iii. If definition is used, we have
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DOI: 10.37394/232020.2024.4.9
Hakan Akkus, Engin Ozkan
E-ISSN: 2732-9941
99
Volume 4, 2024
Thus, we obtain
.
The proofs of the others can be given in the same
way.
In the following theorems, we study the summation
formulas for these hyperbolic quaternions.
Theorem 3.3. Let . We obtain
i.
ii.
.
Proof. i. Using the definition, we have
.
Since
. We get
So, we obtain
.
The proof of the other can be given in the same way.
Theorem 3.4. 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 can be given in the same
way.
Corollary 3.1. We obtain
i.
,
ii.
,
iii.
,
iv.
.
Proof. i. Let is taken in the relation given by
Theorem 3.4. i. We obtain
.
The proofs of the others are shown similarly to i,
using the Theorem 3.4.
Theorem 3.5. (Generating Functions) The
generating functions for hyperbolic -Fibonacci
and -Lucas quaternions are given as follows,
respectively,
and
.
Proof. The following equations are written for the
hyperbolic -Fibonacci sequence.
.
Thus, we have
.
The proof of the other can be given in the same way.
In the following theorems, special generating
functions for these hyperbolic quaternions are
studied. In addition, the Binet formulas are obtained
with the help of generating functions.
Theorem 3.6. For , and , we
obtain
i.
,
ii.
,
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DOI: 10.37394/232020.2024.4.9
Hakan Akkus, Engin Ozkan
E-ISSN: 2732-9941
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Volume 4, 2024
iii.
,
iv.
,
v.
,
vi.
.
Proof. With the help of definitions, Binet formulas
and geometric series, we have
i.
.
Thus, we get
.
The proofs of the others can be given in the same
way.
Theorem 3.7. For
and
quaternions, the
Binet formulas can be obtained with the help of the
generating functions.
Proof. With the help of the roots of the
characteristic equation of these quaternions, the
roots of the equation become
and
. For
quaternions, we obtain
.
Similarly, the Binet formula of the
quaternions
is found.
In the following theorems we calculate special
identities for these hyperbolic quaternions.
Theorem 3.8. (Cassini Identity) For all ,
i.
,
ii.
.
Proof. i. With the Binet formula, we get
(
.
Then, we have
.
So, we obtain
,
The proof of the other can be given in the same way.
Theorem 3.9. (Catalan Identity) For natural
numbers and , we have
i.
,
ii.
.
Proof. i. With the Binet formula, we have
.
So, we can write
.
Thus, we get
.
The proof of the other can be given in the same way.
Theorem 3.10. (Vajda’s Identity) For natural
numbers and , we obtain
i.
,
ii.
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DOI: 10.37394/232020.2024.4.9
Hakan Akkus, Engin Ozkan
E-ISSN: 2732-9941
101
Volume 4, 2024
.
Proof. i. With the Binet formula, we get
)
.
So, we can write
.
Thus, we have
.
The proof of the other can be given in the same way.
Theorem 3.11. (D’ocagne Identity) For natural
numbers , , and , we have
i.
,
ii.
.
Proof. i. Binet formulas are used for proofs. We
obtain
)
.
So, we have
.
The proof of the other can be given in the same way.
In the following theorems we examine the
relationships between these hyperbolic quaternions.
Theorem 3.12. For any integer , we obtain
i.
,
ii.
.
Proof. i. Using the Binet Formula, we have
.
Thus, we obtain
.
The proof of the other can be given in the same way.
Lemma 3.2. We have
i.
,
ii.
,
iii.
,
iv.
.
Proof. i. Using the Binet Formula, we have
.
Thus, we can write
.
The proofs of the others can be given in the same
way.
Theorem 3.13. If natural number and ,
we have
i.
,
ii.
,
iii.
.
Proof. The proofs of the theorem are shown using
the definition, lemma 3.1-3.2, and the Binet
formulas.
In the following theorems, the terms of the -
Fibonacci and -Lucas sequences are associated
with their hyperbolic quaternion values.
Theorem 3.14. For all we get
i.
,
ii.
,
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DOI: 10.37394/232020.2024.4.9
Hakan Akkus, Engin Ozkan
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Volume 4, 2024
iii.
,
iv.
.
Proof. iv. Using the Binet Formula, we have
.
Thus, we obtain
.
The proofs of the others can be given in the same
way.
Theorem 3.15. For all we have
i.
,
ii.
,
iii.
,
iv.
.
Proof. iii. Using the Binet Formula, we have
.
So, we get
The proofs of the others can be given in the same
way.
Theorem 3.16. For all , we obtain
i.
,
ii.
,
iii.
,
iv.
.
Proof. The proofs are shown in the same way as
theorem 3.15.
4 Conclusion
In this study, we defined the hyperbolic -
Jacobsthal and -Jacobsthal-Lucas quaternions.
Then, we obtained some properties of these
quaternions. Also, we examined the relationships
between these quaternions. In addition, we
calculated the special identities of these quaternions.
Moreover, we found the terms of the -
Jacobsthal and -Jacobsthal-Lucas sequences
are associated with their hyperbolic quaternion
values. In the future, we can spread a new approach
to hyperbolic -Jacobsthal and -Jacobsthal-
Lucas octonions and sedenions.
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PROOF
DOI: 10.37394/232020.2024.4.9
Hakan Akkus, Engin Ozkan
E-ISSN: 2732-9941
104
Volume 4, 2024
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
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.
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DOI: 10.37394/232020.2024.4.9
Hakan Akkus, Engin Ozkan
E-ISSN: 2732-9941
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