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1. UtilitasMathematica
ISSN 0315-3681 Volume 120, 2023
1
ON SOME K- ORESME HYBRID NUMBERS
Serpil HALICI1, Elifcan SAYIN2
1,2
Pamukkale Uni., Science Fac., Depart. of Math., Denizli, Turkey
shalici@pau.edu.tr1
, elifcan7898@gmail.com2
Abstract
The main aim of this study is to define k- Oresme Hybrid numbers by using Oresme numbers and
to investigate some of their algebraic properties and applications. For this purpose, many
formulas and relations have been created for these new numbers, including some implicit
summation formulas, and links with the other numbers. Moreover, the fundamental properties in
terms of integer sequences are also given.
Keywords: Oresme numbers, Hybrid numbers, Recurrence relations.
1. Introduction
Sequences with the recurence relations have an increasing interest and usage in number theory.
Such sequences are used not only in number theory, but also in important fields such as physics,
biology, cryptography and computers, and nowadays the interest in these sequences is increasing.
The most widely used of these sequences is the Horadam sequence. The Horadam sequence with
arbitrary initial values π€0 and π€1 is defined with the help of the second-order linear recurrence
relation { π€π = π€π(π, π; π, π)} πβ₯0
π€π = ππ€πβ1 β ππ€πβ2. (1.1)
Based on the initial values, the parameters π and π it is possible to obtain a large number of
different sequences from this sequence. Information about these sequences can be found in the
references [1, 6, 7, 10]. Oresme is known as the first person to define and investigate the topic of
fractional powers [11]. While examining rational number sequences and their sums, Oresme
inadvertently encountered a special case of the Horadam sequence, but this work was not
published then was later reworked by Horadam. To put it briefly, Horadam first discussed the
extension of arbitrary parameters π and π to rational numbers in 1974[8]. In his study, the author
obtained the relationship between these numbers and Fibonacci numbers by giving the number e
which characterizes the Oresme numbers. This character numberπ = πππ β ππ2
β π2
obtained
by Horadam has been observed and used in many important identities involving Oresme
numbers, such as the Cassini identity. The distinguishing properties of Oresme numbers stem
from the roots of the characteristic equation of these numbers, and therefore they are considered
as a degenerate subsequence of the Horadam sequence.
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Oresme sequence, which are obtained by taking the values π, π, π, π in the Horadam sequence as
0,
1
2
, 1,
1
4
, respectively, are found by the second order linear recurrence relation.
ππ+2 = ππ+1 β
1
4
ππ (1.2)
Some elements of this sequence are
{β¦ , β64, β24, β8, β2, 0,
1
2
,
1
2
,
3
8
,
4
16
, β― }.
In [2], Morales investigated Oresme polynomials and their derivatives. He defined the π- Oresme
sequence {ππ
π} πβ₯2, which is a type of generalization of Oresme numbers by
ππ
(π)
= ππβ1
(π)
β
1
π2
ππβ2
(π)
(1.3)
with π1
π
=
1
π
, π0
π
= 0. For π = 2, this sequence is reduced to the classical Oresme sequence.
For π2
β 4 > 0, the Binet's formula, which gives these numbers, can be given by using the
recurrence relation .
ππ
(π)
=
1
βπ2β4
[(
π+βπ2β4
2π
)
π
β (
πββπ2β4
2π
)
π
] (1.4)
Also, Morales discussed the matrix forms of Oresme polynomials by extending the k- Oresme
sequence to rational functions and investigated the relationships between Oresme polynomials
and their derivatives. Recently, there are some studies on Oresme numbers and their applications.
For these studies, the references can be examined in [2, 3, 4, 5, 8, 13, 17]. In [4], Halici et al.
examined k- Oresme numbers involving negative indices. Recursive relation of this sequence is
denoted by {πβπ
(π)
}
πβ₯0
and defined by
πβπ
(π)
= π2
(πβπ+1
(π)
β πβπ+2
(π)
). (1.5)
The π π‘β term of this sequence is
πβπ
(π)
= βπ2π (πΌπβπ½π)
βπ2β4
(1.6)
with initial conditions πβ1
(π)
= βπ , π0
(π)
= 0. The values πΌ and π½ are as in equation (1.4).
2. π -ORESME HYBRID NUMBERS
Ozdemir defined a non-commutative number system in 2018 and called it hybrid numbers. The
author examined in detail the algebraic and geometrical properties of this new number system,
which he described in this study[12]. In 2018, Szynal-Liana introduced Horadam Hybrid
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numbers and examined their special cases[14]. In 2019, Szynal-Liana considered Fibonacci
Hybrid numbers and examined them[15]. In 2019, Szynal-Liana examined Jacobsthal and
Jacobsthal-Lucas hybrid numbers[16]. In 2021, IΕbilir Z. and Gurses N. examined Pentanacci
and Pentanacci Lucas hybrid numbers[9].
Hybrid numbers and Horadam Hybrid numbers, respectively, are defined by
π = {π§ = π + ππ’ + ππ + ππ‘ βΆ π, π, π, π β β}
and
π»π = π
π + ππ+1π’ + ππ+2π + ππ+3π‘
The relations provided between the three different base elements used in the set π are
π π
= β1, πΊ2
= 0, π2
= 1 , ππ = βππ = πΊ + π
The conjugate of any hybrid number π§ in the π set is defined by
π§Μ = π β ππ’ β ππ β ππ‘ ,
The character value of element π§ of π is
π(π§) = π§π§Μ = π§Μ π§ = π2
+ (π β π)2
β π2
β π2
This character number is commonly used to determine the generalized norm of a hybrid number.
We suggest you look at Ozdemir's work to get detailed and useful information about hybrid
numbers [12]. This work has inspired many studies and applications involving these numbers.
One of these studies, Oresme Hybrid numbers, was discussed by Syznal et al.(See, [17]). In [17],
the authors defined the Oresme Hybrid number for positive number π as follows.
ππ»π = ππ + ππ+1π’ + ππ+2π + ππ+3π‘ (1.7)
In addition, the authors introduced the π-Oresme Hybrid number.
ππ»π
(π)
= ππ
(π)
+ ππ+1
(π)
π’ + ππ+2
(π)
π + ππ+3
(π)
π‘ (1.8)
Also, in the same study some basic identities including π- Oresme Hybrid numbers are given
with the help of iterative relation, but other important identities are not given (see, [17]. Thr. 2.4).
They also defined Oresme Hybrationals for a nonzero real variable π₯ .
ππ»π(π₯) = ππ(π₯) + ππ+1(π₯)π’ + ππ+2(π₯)π + ππ+3(π₯)π‘ (1.9)
with π β₯ 0.
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In this current study, we introduced and investigated π- Oresme Hybrid numbers ππ»π
(π)
in detail
by making use of all the studies on rational functions. For π β₯ 2, we consider some fundamental
and important identities involving the numbers ππ»π
(π)
and having applications in the literature.
Firstly, we give the Binet formula which provides the derivation of many important equations.
Theorem 1. Binet formula for π-Oresme Hybrid numbers is
ππ»π
(π)
=
1
βπ2β4
(πΌπ
πΌ
Μ β π½π
π½
Μ) (1.10)
where,
πΌ
Μ = 1 + πΌπ’ + πΌ2
π + πΌπ
π‘, π½
Μ = 1 + π½π’ + π½2
π + π½π
π‘. (1.11)
Proof. We use the equality ππ»π
(π)
= ππ
(π)
+ ππ+1
(π)
π’ + ππ+2
(π)
π + ππ+3
(π)
π‘. Let's substitute every
term in this equation. Then, we get
ππ»π
(π)
=
1
βπ2 β 4
[πΌπ(1 + πΌπ’ + πΌ2
π + πΌπ
π‘) + π½π(1 + π½π’ + π½2
π + π½π
π‘)].
If we also use the equations (1.11), then we obtain
ππ»π
(π)
=
1
βπ2 β 4
(πΌπ
πΌ
Μ β π½π
π½
Μ)
which indicates that the proof is complete.
Recurrence relation giving the π- Oresme Hybrid numbers is
ππ»π+2
(π)
= ππ»π+1
(π)
β
1
π2
ππ»π
(π)
(1.12)
with ππ»0
(π)
=
1
π
(π’ + π +
π2β1
π2
π‘) and ππ»1
(π)
=
1
π
(1 + π’ +
π2β1
π2
π +
π2β2
π2
π‘). The correctness
of this relation can be seen using the ππ‘β term of the sequence and induction.
In the following theorem, we give the generating function of the sequence involving numbers
ππ»π
(π)
.
Theorem 2. For the numbers ππ»π
(π)
, the following equation is satisfied.
5. UtilitasMathematica
ISSN 0315-3681 Volume 120, 2023
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β ππ»π
(π)
π§π
πβ₯0 =
ππ»0
(π)
(1βπ§)+π§ππ»1
(π)
1βπ§+
π§2
π2
(1.13)
Proof. From the definition of the generating function,
π(π§) = ππ»0
(π)
+ ππ»1
(π)
π§ + ππ»2
(π)
π§2
+ ππ»3
(π)
π§3
β¦
βπ§π(π§) = βπ§ππ»0
(π)
β π§2
ππ»1
(π)
β π§3
ππ»2
(π)
β π§4
ππ»3
(π)
+ β―
π§2
π2
π(π§) =
π§2
π2
ππ»0
(π)
+
π§3
π2
ππ»1
(π)
+
π§4
π2
ππ»2
(π)
+
π§5
π2
ππ»3
(π)
+ β―
can be written. If necessary operations are made on the last three equations, then the function
π(π§) (1 β π§ +
π§2
π2
) is
ππ»0
(π)
+ π§(ππ»1
(π)
β ππ»0
(π)
) + π§2
(ππ»2
(π)
β ππ»1
(π)
+
1
π2
ππ»0
(π)
) + π§3
(ππ»3
(π)
β ππ»2
(π)
+
1
π2
ππ»1
(π)
) + β―.
So, by making necessary simplifications and calculations on the last equation, the following
equation is obtained
π(π§) =
ππ»0
(π)
(1 β π§) + π§ππ»1
(π)
1 β π§ +
π§2
π2
which indicates that the desired equality is true. Thus, the proof is completed.
In the following theorem, we give the Cassini identity containing π- Oresme Hybrid numbers.
Theorem 3. The elements of the sequence {ππ»π
(π)
}
πβ₯0
satisfy the following identity.
ππ»π+1
(π)
ππ»πβ1
(π)
β (ππ»π
(π)
)
2
= β (
1
π
)
2π
πΌ
Μπ½
Μ . (1.14)
Proof. Using ππ»π+1
(π)
=
1
βπ2β4
(πΌπ+1
πΌ
Μ β π½π+1
π½
Μ), ππ»πβ1
(π)
=
1
βπ2β4
(πΌπβ1
πΌ
Μ β π½πβ1
π½
Μ) and
(ππ»π
(π)
)
2
=
1
π2β4
[(πΌπ
πΌ
Μ)2
β 2(πΌπ½)π
πΌ
Μπ½
Μ + (π½π
π½
Μ)
2
], then the left-hand side of (1.14) will be
as follows.
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πΏπ»π =
1
π2 β 4
[(πΌπ+1
πΌ
Μ β π½π+1
π½
Μ)(πΌπβ1
πΌ
Μ β π½πβ1
π½
Μ) β ((πΌπ
πΌ
Μ)2
β 2(πΌπ½)π
πΌ
Μπ½
Μ + (π½π
π½
Μ)
2
)]
and
ππ»π+1
(π)
ππ»πβ1
(π)
β (ππ»π
(π)
)
2
= β
1
π2β4
(πΌπ½)π
πΌ
Μπ½
Μ [
πΌ2+π½2
πΌπ½
β 2].
Here, if we also use the relations between the roots, then we get the desired result. That is, we
have
ππ»π+1
(π)
ππ»πβ1
(π)
β (ππ»π
(π)
)
2
= β (
1
π
)
2π
πΌ
Μπ½
Μ .
Thus, the theorem is completed.
In the following theorem, we give the Catalan identity provided by the π- Oresme Hybrid
numbers.
Theorem 4. For π β₯ π and π β₯ 2, the following equality is true.
ππ»π+π
(π)
ππ»πβπ
(π)
β (ππ»π
(π)
)
2
= β(π)β2π+π
πΌ
Μπ½
Μ(ππ
(π)
)
2
. (1.15)
Proof. If we use the Binet formula to write the numbers ππ»π+π
(π)
, ππ»πβπ
(π)
and ππ»π
(π)
, then left-
hand side of the equation to be proved is
[(πΌπ
πΌ
Μ)2
+ 2(πΌπ½)π
πΌ
Μπ½
Μ + (π½π
π½
Μ)
2
β (πΌπ½)π
πΌ
Μπ½
Μ (
πΌ
π½
)
π
β (πΌπ½)π
πΌ
Μπ½
Μ (
π½
πΌ
)
π
β (πΌπ
πΌ
Μ)2
β (π½π
π½
Μ)
2
].
On the last equation, if necessary simplifications and algebraic operations are completed, then
ππ»π+π
(π)
ππ»πβπ
(π)
β (ππ»π
(π)
)
2
= β(π)β2π+π
πΌ
Μπ½
Μ(ππ
(π)
)
2
is obtained that the claim is proven.
If we pay attention here, in the situation π = 1 this equation is reduced to the Cassini identity for
the π- Oresme Hybrid numbers.
In the following theorem, we give the d'Ocagne identity, which includes π- Oresme Hybrid
numbers.
Theorem 5. For positive integers π, π the following equation is true.
ππ»π+1
(π)
ππ»π
(π)
β ππ»π
(π)
ππ»π+1
(π)
= (
1
π
)
2πβ1
πΌ
Μπ½
Μππβπ
(π)
. (1.16)
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Proof. Let's take the values on the left side of the equation that needs to be proven separately.
ππ»π+1
(π)
ππ»π
(π)
=
1
π2 β 4
[πΌπ+π+1(πΌ
Μ)2
β πΌπ+1
π½π
πΌ
Μπ½
Μ β π½π+1
πΌπ
πΌ
Μπ½
Μ + π½π+π+1
(π½
Μ)
2
].
ππ»π
(π)
ππ»π+1
(π)
=
1
π2 β 4
[πΌπ+π+1(πΌ
Μ)2
+ π½π+π+1
(π½
Μ)
2
β πΌπ+1
π½π
πΌ
Μπ½
Μ β π½π+1
πΌπ
πΌ
Μπ½
Μ].
From the last two equations, we get
ππ»π+1
(π)
ππ»π
(π)
β ππ»π
(π)
ππ»π+1
(π)
=
1
π2β4
(πΌ
Μπ½
Μ)[πΌπ
π½π(πΌ β π½) + πΌπ
π½π(π½ β πΌ)],
ππ»π+1
(π)
ππ»π
(π)
β ππ»π
(π)
ππ»π+1
(π)
=
1
π2β4
(πΌ
Μπ½
Μ) (
βπ2β4
π
) (
1
π
)
2π
(πΌπβπ
β π½πβπ),
ππ»π+1
(π)
ππ»π
(π)
β ππ»π
(π)
ππ»π+1
(π)
=
(πΌ
Μπ½
Μ)(
βπ2β4
π
)(
1
π
)
2π
βπ2β4
(πΌπβπβπ½πβπ)
βπ2β4
,
ππ»π+1
(π)
ππ»π
(π)
β ππ»π
(π)
ππ»π+1
(π)
= (
1
π
)
2πβ1
πΌ
Μπ½
Μππβπ
(π)
.
Thus, the claim is true.
In the following theorem we give the Honsberger's identity which includes the π- Oresme Hybrid
numbers.
Theorem 6. For positive integers π, π the following equation is satisfied.
ππ»π+π
(π)
= πππ
(π)
ππ»π+1
(π)
β
1
π
ππβ1
(π)
ππ»π
(π)
. (1.17)
Proof. To see that the equation in (1.17) is true, we write the value ππ»π+π
(π)
as follows.
ππ»π+π
(π)
= π1ππ
(π)
+ π2ππβ1
(π)
(1.18)
Let's see that the values π1 and π2 are π1 = πππ»π+1
(π)
and π2 = β
1
π
ππ»π
(π)
.
If we substitute the Binet formula in equation (1.18) and do the necessary operations, then we get
the following equation.
πΌπ+π
πΌ
Μ β π½π+π
π½
Μ = π1πΌπ
β π1π½π
+ π2πΌπβ1
β π2π½πβ1
= πΌπβ1(π1πΌ + π2) β π2π½πβ1(π1π½ +
π2).
From the last equality, we can write
πΌ
ΜπΌπ+1
= π1πΌ + π2 (1.19)
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π½
Μπ½π+1
= π1π½ + π2 (1.20)
By the aid of (1.19) and (1.20) equations, we obtain
π1 =
πΌ
ΜπΌπ+1βπ½
Μπ½π+1
πΌβπ½
= πππ»π+1
(π)
(1.21)
π2 =
πΌ
ΜπΌπβπ½
Μπ½π
πΌβπ½
= β
1
π
ππ»π
(π)
(1.22)
If we substitute these values π1 and π2 in the equation (1.18), then we get
ππ»π+π
(π)
= πππ
(π)
ππ»π+1
(π)
β
1
π
ππβ1
(π)
ππ»π
(π)
which is desired result. Thus, the proof is completed.
Now, we give a formula that gives the sum of the first π of the π- Oresme Hybrid numbers in the
theorem below.
Theorem 7. For the π-Oresme Hybrid numbers, the following equation is satisfied.
β ππ»π
(π)
= π2
(ππ»1
(π)
β ππ»π+2
(π)
).
π
π=1 (1.23)
Proof. By using Binet's formula,
β ππ»π
(π)
=
π
π=1
1
βπ2 β 4
[πΌ
Μ (
1 β πΌπ+1
1 β πΌ
) β π½
Μ (
1 β π½π+1
1 β π½
)]
can be written. Also, if the relations between πΌ and π½ are used, then
β ππ»π
(π)
π
π=1
=
1
βπ2 β 4
[πΌ
Μ β πΌ
Μπ½
Μ β πΌ
ΜπΌπ+1
+ πΌ
Μπ½πΌπ+1
β π½
Μ + π½
ΜπΌ + π½
Μπ½π+1
β πΌπ½
Μπ½π+1
],
β ππ»π
(π)
π
π=1
=
1
βπ2 β 4
[
βπΌ
ΜπΌπ+2
+ π½
Μπ½π+2
+ πΌπΌ
Μ β π½π½
Μ
πΌπ½
],
β ππ»π
(π)
= π2
(ππ»1
(π)
β ππ»π+2
(π)
) ,
π
π=1
is obtained. Thus, the proof is completed.
In the following theorem, we obtained the formula gives the sum of squares of the π- Oresme
Hybrid numbers.
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9
Theorem 8. The sum of squares of the π- Oresme Hybrid numbers is
β (ππ»π
(π)
)
2
=
βπ4(ππ»1
(π)
+(ππ»π‘+1
(π)
)
2
)+ππ»0
(π)
β(ππ»π‘
(π)
)
2
β2(
1βπ₯π‘
1βπ₯
)πΌ
Μπ½
Μ
β1β2π2
π‘
π=1 . (1.24)
Proof. Let's denote the sum of the squares with the letter T.
(ππ»1
(π)
)
2
+ (ππ»2
(π)
)
2
+ (ππ»3
(π)
)
2
+ β― = π
For each term, the recurrence relation of the π- Oresme Hybrid numbers is written and the
necessary calculations are made for each, and the following equation is obtained.
ππ»π
(π)
= ππ»π+1
(π)
+
1
π2
ππ»πβ1
(π)
π = β(ππ»π
(π)
)
2
= β [
π2
ππ»π+1
(π)
+ ππ»πβ1
(π)
π2
]
2
,
π‘
π=1
π‘
π=1
π4
π = π4
β(ππ»π+1
(π)
)
2
+ β(ππ»πβ1
(π)
)
2
+ 2π2
β ππ»π+1
(π)
ππ»πβ1
(π)
π‘
π=1
π‘
π=1
π‘
π=1
.
Here, let's also use the Cassini identity, that is, ππ»π+1
(π)
ππ»πβ1
(π)
= (ππ»π
(π)
)
2
β (
1
π
)
2π
πΌ
Μπ½
Μ.
Then, we get
π4
π = π4
(π β (ππ»1
(π)
)
2
+ (ππ»π‘+1
(π)
)
2
) + (π + (ππ»0
(π)
)
2
β (ππ»π‘
(π)
)
2
) +
2π2 β (ππ»π
(π)
)
2
β (
1
π
)
2π
πΌ
Μπ½
Μ
π‘
π=1 ,
π4
π = π4
(π β (ππ»1
(π)
)
2
+ (ππ»π‘+1
(π)
)
2
) + (π + (ππ»0
(π)
)
2
β (ππ»π‘
(π)
)
2
) + 2π2
π β
2 β (π)β2π+2
πΌ
Μπ½
Μ
π‘
π=1 ,
π4
π = π(π4
+ 2π2
+ 1) β π4
(ππ»1
(π)
)
2
+ π4
(ππ»π‘+1
(π)
)
2
+ (ππ»0
(π)
)
2
β (ππ»π‘
(π)
)
2
β 2
1βπ₯π‘
1βπ₯
πΌ
Μπ½
Μ.
Thus, we obtain
π = β (ππ»π
(π)
)
2
=
βπ4(ππ»1
(π)
+(ππ»π‘+1
(π)
)
2
)+ππ»0
(π)
β(ππ»π‘
(π)
)
2
β2(
1βπ₯π‘
1βπ₯
)πΌ
Μπ½
Μ
β1β2π2
π‘
π=1 .
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In the following theorem, we give the formula of sums of π- Oresme Hybrid numbers with even
and odd indices terms.
Theorem 9. The sums of π- Oresme Hybrid numbers with even and odd indices terms are
π) β ππ»2π
(π)
π
π=1 = π2(ππ»0 β ππ»2π+2) β (ππ»β1 β ππ»2π+1). (1.25)
ππ) β ππ»2π+1
(π)
π
π=1 = π2(ππ»0 β ππ»2π+1) β (ππ»β1 β ππ»2π). (1.26)
Proof. i) From the Binet formula of π- Oresme Hybrid numbers,
β ππ»2π
(π)
π
π=1
=
1
βπ2 β 4
[πΌ
Μ β πΌ2π
β
π
π=1
π½
Μ β π½2π
π
π=1
],
β ππ»2π
(π)
π
π=1
=
1
βπ2 β 4
[πΌ
Μ (
1 β πΌ2π+2
1 β πΌ
) β π½
Μ (
1 β π½2π+2
1 β π½
)],
is written. By using the relationships πΌ + π½ = 1, πΌ β π½ =
βπ2β4
π
, πΌπ½ =
1
π2
β ππ»2π
(π)
π
π=1
=
1
βπ2 β 4
[
πΌ
Μ β π½
Μ β (πΌ
Μπ½ β π½
ΜπΌ) β (πΌ
Μπ½πΌ2π+2
β π½
ΜπΌπ½2π+2
) β (πΌ
ΜπΌ2π+2
β π½
Μπ½2π+2
)
πΌπ½
],
β ππ»2π
(π)
π
π=1
=
π2
(πΌ
Μ β π½
Μ)
βπ2 β 4
β
(πΌ
ΜπΌβ1
β π½
Μπ½β1
)
βπ2 β 4
β
(πΌ
ΜπΌ2π+1
β π½
Μπ½2π+1
)
βπ2 β 4
β
π2
(πΌ
ΜπΌ2π+2
β π½
Μπ½2π+2
)
βπ2 β 4
,
β ππ»2π
(π)
π
π=1
= π2
(ππ»0
(π)
β ππ»2π+2
(π)
) β (ππ»β1
(π)
β ππ»2π+1
(π)
),
is obtained. The sum of odd terms of π- Oresme Hybrid numbers can also be calculated as in the
proof of even terms.
Thus, the proof is completed.
3. Conclusion
In this study, k- Oresme Hybrid numbers are defined and examined. Some of their algebraic
properties and applications are introduced. Some useful sum formulas for these newly defined
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ISSN 0315-3681 Volume 120, 2023
11
numbers are given. In addition, many basic and important identities and equations are obtained
by considering the relations between the numbers studied here and Horadam numbers.
References
[1] Cerda, G. (2012). Matrix methods in Horadam sequences.BoletΓn de MatemΓ‘ticas, 19(2), 97-
106.
[2] Cerda-Morales,G. (2019). Oresme polynomials and their derivatives. arxiv preprint
arXiv:1904.01165.
[3] Cook, C.K.: Some sums related to sums of Oresme numbers, in: Applications of Fibonacci
Numbers, vol.9, Proceedings of the Tenth International Research Conference on Fibonacci
Numbers and their Applications, Kluwer Academic Publishers, 2004, pp.87β99.
[4] HalΔ±cΔ±, S., SayΔ±n, E., Gur, Z. B. (2022). On k-Oresme Numbers with Negative Indices.
Engineering (ICMASE 2022), 4-7 July 2022, Technical University of Civil Engineering
Bucharest, Romania, 56.
[5] HalΔ±cΔ±, S., Gur, Z. B. On Some Derivatives of k- Oresme Polynomials, Bulletin of
International Mathematical Virtual Institute, (accepted).
[6] Horadam, A. F. (1965). Basic properties of a certain generalized sequence of numbers. The
Fibonacci Quarterly, 3(3), 161-176.
[7] Horadam, A. F. (1967). Special properties of the sequence γ wγ_n=w_n (a,b;p,q).
Fibonacci Quarterly, 5(5), 424-434.
[8] Horadam, A. F. (1974). Oresme numbers. Fibonacci Quarterly, 12(3), 267-271.
[9] IΕbilir, Z., GΓΌrses, N. (2021). Pentanacci and Pentanacci-Lucas hybrid numbers. Journal of
Discrete Mathematical Sciences and Cryptography, 1-20.
[10] Larcombe, P. J. (2017).Horadam sequences: A surve update and extension. Bulletin of the
Institute of Combinatorics and its Applications, Volume 80, pp.99-118.
[11] Oresme, N. (1961). Quaestiones super geometriam Euclidis (Vol. 3). Brill Archive.
[12] Ozdemir, M. (2018). Introduction to hybrid numbers. Advances in applied Clifford algebras,
28(1), 1-32.
[13] Senturk, G. Y., Yuce, N. G. S. (2021). A New Look on Oresme Numbers: Dual-Generalized
Complex Component Extension. In Conference Proceedings of Science and (Vol. 4, No. 2,
pp. 206-214).
[14] Szynal-Liana, A. (2018). The Horadam Hybrid Numbers. Discussiones Mathematicae
General Algebra and Applications 38 (2018), 91β98.
[15] Szynal-Liana, A., Wioch, I. (2019). The Fibonacci hybrid numbers. Utilitas Mathematica,
Vol. 110 (2019).
[16] Szynal-Liana, A., WΕoch, I. (2019). On Jacobsthal and Jacobsthal-Lucas hybrid numbers. In
Annales Mathematicae Silesianae (Vol. 33, pp. 276-283).
[17] Szynal-Liana, A., Wloch, I. (2024). Oresme Hybrid Numbers and Hybrationals. Kragujevac
Journal of Mathematics, 48(5), 747-753.