Bu tez bes bolumden olusmaktadir. Birinci bolumde tezin onemi irdelenmistir. Ikinci bolumde, tezd... more Bu tez bes bolumden olusmaktadir. Birinci bolumde tezin onemi irdelenmistir. Ikinci bolumde, tezde kullanilan temel tanim ve kavramlar verilmistir. Ucuncu bolumde, helisel vektor alanlari ve bunlarin dual kuaterniyonlarla iliskisi ele alinmistir. Dorduncu bolumde, Oklid uzayinda helisel vektor alanlari yardimiyla bir parametreli hareketlerin integral egrilerinin cinsi belirlenmistir. Son bolumde, Oklid uzayinda yapilan islemler Lorenz uzayina genellestirilmistir.AbstractThis thesis consists of five chapters.In the first chapter, is given the importance of thesisThe second chapter, is devoted to the introduction.In the third chapter, the relationship between helicoidal vector fields and Dualquaternions is examined.In the fourth chapter, in Euclidean space, the classification of the integral curves ofthe one parameter motions are given by the help of the helicoidal vector fields.In the last chapter, the results which are required in Euclidean space are generalizedinto Lorentzian space.
Fibonacci and Lucas numbers have been the most popular integer sequences since they were defined.... more Fibonacci and Lucas numbers have been the most popular integer sequences since they were defined. These integer sequences have many uses, from nature to computer science, from art to financial analysis. Many researchers have worked on this subject. Sedenions form a 16-dimensional algebra on the field of real numbers. Various systems can be constructed by using the terms of special integer sequences instead of terms in sedenions. In this study, we define dual Fibonacci (DFS) and dual Lucas sedenions (DLS) with the help of Fibonacci and Lucas termed sedenions. Then we calculate some special identities for DFS and DLS such as Vajda's, Catalan's, d'Ocagne's, Cassini's.
Turkish Journal of Mathematics and Computer Science, 2021
In this study, we gave a generalization on Pell and Pell-Lucas octonions over the algebra O(a, b,... more In this study, we gave a generalization on Pell and Pell-Lucas octonions over the algebra O(a, b, c) where a, b and c are real numbers. For these number sequences, we obtain Binet formulas and gave some wellknown identities such as Catalan's identity, Cassini's identity and d'Ocagne's identity.
We investigate the properties of $k-$Fibonacci and $k-$Lucas quaternions over the generalized qua... more We investigate the properties of $k-$Fibonacci and $k-$Lucas quaternions over the generalized quaternion algebra. After presenting generating functions and Binet's formulas for these types of quaternions, we calculate several well-known identities such as Catalan's, Cassini's and d'Ocagne's identities for $k-$Fibonacci and $k-$Lucas generalized quaternions.
The sedenions form a 16-dimensional non-associative and non-commutative algebra over the set of r... more The sedenions form a 16-dimensional non-associative and non-commutative algebra over the set of real numbers. In this paper, we introduce the Fibonacci and Lucas sedenions. We present generating functions and Binet formulas for the Fibonacci and Lucas sedenions, and derive adaptations for some well-known identities of Fibonacci and Lucas numbers.
We study on Jacobsthal and Jacobsthal-Lucas generalized octonions over the algebra O(a,b,c) where... more We study on Jacobsthal and Jacobsthal-Lucas generalized octonions over the algebra O(a,b,c) where a,b and c are real numbers. We present Binet formulas for these types of octonions. Furthermore, we give some well-known identities such as Catalan's, Cassini's, d'Ocagne's identities and other special identities for Jacobsthal and Jacobsthal-Lucas generalized octonions.
Hacettepe Journal of Mathematics and Statistics, 2020
The hyperbolic Fibonacci and hyperbolic Lucas functions have been introduced before and have been... more The hyperbolic Fibonacci and hyperbolic Lucas functions have been introduced before and have been improved to functions of the symmetrical form. In this paper, we generalize the mentioned definitions, which will be called Horadam hyperbolic sine function (HSF) and Horadam hyperbolic cosine function (HCF). Further, we present many identities and hyperbolic properties of our new definitions.
In this paper, oscillation criteria are obtained for higher-order neutral-type nonlinear delay di... more In this paper, oscillation criteria are obtained for higher-order neutral-type nonlinear delay difference equations of the form ∆(rn(∆ k−1 (yn + pnyτ n)) + qnf (yσ n) = 0, n ≥ n0, (0.1) where rn, pn, qn ∈ [n0, ∞), rn > 0 , qn > 0 ; 0 ≤ pn ≤ p0 < ∞ ; lim n→∞ τn = ∞ , lim n→∞ σn = ∞ ; σn ≤ n , σn is nondecreasing; ∆τn ≥ τ0 > 0 ; τσ = στ ; f (u) u ≥ m > 0 for u ̸ = 0. Moreover, we provide some examples to illustrate our main results.
In this paper, we study the statistical approximation properties of real and complex Post-Widder ... more In this paper, we study the statistical approximation properties of real and complex Post-Widder operators based on q-integers. We also obtain a Voronovskaya-type formula in statistical sense for these operators.
The aim of this work is to introduce split Pell and split Pell-Lucas quaternions. We give generat... more The aim of this work is to introduce split Pell and split Pell-Lucas quaternions. We give generating functions and Binet formulas for these numbers. Also, we obtain many identities for split Pell and split Pell-Lucas quaternions including Catalan's identity, Cassini's identity and d'Ocagne's identity.
Halici (Adv Appl Clifford Algebr 25(4):905–914, 2015) defined dual Fibonacci and dual Lucas octon... more Halici (Adv Appl Clifford Algebr 25(4):905–914, 2015) defined dual Fibonacci and dual Lucas octonions by the relations $${\widetilde{Q}_{n}={Q}_n+\varepsilon Q_{n+1}}$$Q~n=Qn+εQn+1 and $${\widetilde{P}_n=P_n+\varepsilon P_{n+1}}$$P~n=Pn+εPn+1 for every integer n where $${Q_n}$$Qn and $${P_n}$$Pn are the Fibonacci and Lucas octonions respectively, and $${\varepsilon}$$ε is the dual unit. The aim of this paper is to investigate properties of dual Fibonacci and dual Lucas octonions. After obtaining the Binet formulas for the sequences $${\{\widetilde{Q}_n \}_{n=0}^\infty}$${Q~n}n=0∞ and $${\{\widetilde{P}_n \}_{n=0}^\infty}$${P~n}n=0∞, we derive some identities for these sequences such as Catalan’s, Cassini’s and d’Ocagne’s identities.
In this article, we give the most general form of the quaternions algebra depending on 3-paramete... more In this article, we give the most general form of the quaternions algebra depending on 3-parameters. We define 3-parameter generalized quaternions (3PGQs) and study on various properties and applications. Firstly we present the definiton, the multiplication table and other properties of 3PGQs such as addition-substraction, multiplication and multiplication by scalar operations, unit and inverse elements, conjugate and norm. We give matrix representation and Hamilton operators for 3PGQs.We get polar represenation, De Moivre's and Euler's formulas with the matrix representations for 3PGQs. Besides, we give relations among the powers of the matrices associated with 3PGQs. Finally, Lie group and Lie algebra are studied and their matrix representations are shown. Also the Lie multiplication and the killing bilinear form are given.
Bu tez bes bolumden olusmaktadir. Birinci bolumde tezin onemi irdelenmistir. Ikinci bolumde, tezd... more Bu tez bes bolumden olusmaktadir. Birinci bolumde tezin onemi irdelenmistir. Ikinci bolumde, tezde kullanilan temel tanim ve kavramlar verilmistir. Ucuncu bolumde, helisel vektor alanlari ve bunlarin dual kuaterniyonlarla iliskisi ele alinmistir. Dorduncu bolumde, Oklid uzayinda helisel vektor alanlari yardimiyla bir parametreli hareketlerin integral egrilerinin cinsi belirlenmistir. Son bolumde, Oklid uzayinda yapilan islemler Lorenz uzayina genellestirilmistir.AbstractThis thesis consists of five chapters.In the first chapter, is given the importance of thesisThe second chapter, is devoted to the introduction.In the third chapter, the relationship between helicoidal vector fields and Dualquaternions is examined.In the fourth chapter, in Euclidean space, the classification of the integral curves ofthe one parameter motions are given by the help of the helicoidal vector fields.In the last chapter, the results which are required in Euclidean space are generalizedinto Lorentzian space.
Fibonacci and Lucas numbers have been the most popular integer sequences since they were defined.... more Fibonacci and Lucas numbers have been the most popular integer sequences since they were defined. These integer sequences have many uses, from nature to computer science, from art to financial analysis. Many researchers have worked on this subject. Sedenions form a 16-dimensional algebra on the field of real numbers. Various systems can be constructed by using the terms of special integer sequences instead of terms in sedenions. In this study, we define dual Fibonacci (DFS) and dual Lucas sedenions (DLS) with the help of Fibonacci and Lucas termed sedenions. Then we calculate some special identities for DFS and DLS such as Vajda's, Catalan's, d'Ocagne's, Cassini's.
Turkish Journal of Mathematics and Computer Science, 2021
In this study, we gave a generalization on Pell and Pell-Lucas octonions over the algebra O(a, b,... more In this study, we gave a generalization on Pell and Pell-Lucas octonions over the algebra O(a, b, c) where a, b and c are real numbers. For these number sequences, we obtain Binet formulas and gave some wellknown identities such as Catalan's identity, Cassini's identity and d'Ocagne's identity.
We investigate the properties of $k-$Fibonacci and $k-$Lucas quaternions over the generalized qua... more We investigate the properties of $k-$Fibonacci and $k-$Lucas quaternions over the generalized quaternion algebra. After presenting generating functions and Binet's formulas for these types of quaternions, we calculate several well-known identities such as Catalan's, Cassini's and d'Ocagne's identities for $k-$Fibonacci and $k-$Lucas generalized quaternions.
The sedenions form a 16-dimensional non-associative and non-commutative algebra over the set of r... more The sedenions form a 16-dimensional non-associative and non-commutative algebra over the set of real numbers. In this paper, we introduce the Fibonacci and Lucas sedenions. We present generating functions and Binet formulas for the Fibonacci and Lucas sedenions, and derive adaptations for some well-known identities of Fibonacci and Lucas numbers.
We study on Jacobsthal and Jacobsthal-Lucas generalized octonions over the algebra O(a,b,c) where... more We study on Jacobsthal and Jacobsthal-Lucas generalized octonions over the algebra O(a,b,c) where a,b and c are real numbers. We present Binet formulas for these types of octonions. Furthermore, we give some well-known identities such as Catalan's, Cassini's, d'Ocagne's identities and other special identities for Jacobsthal and Jacobsthal-Lucas generalized octonions.
Hacettepe Journal of Mathematics and Statistics, 2020
The hyperbolic Fibonacci and hyperbolic Lucas functions have been introduced before and have been... more The hyperbolic Fibonacci and hyperbolic Lucas functions have been introduced before and have been improved to functions of the symmetrical form. In this paper, we generalize the mentioned definitions, which will be called Horadam hyperbolic sine function (HSF) and Horadam hyperbolic cosine function (HCF). Further, we present many identities and hyperbolic properties of our new definitions.
In this paper, oscillation criteria are obtained for higher-order neutral-type nonlinear delay di... more In this paper, oscillation criteria are obtained for higher-order neutral-type nonlinear delay difference equations of the form ∆(rn(∆ k−1 (yn + pnyτ n)) + qnf (yσ n) = 0, n ≥ n0, (0.1) where rn, pn, qn ∈ [n0, ∞), rn > 0 , qn > 0 ; 0 ≤ pn ≤ p0 < ∞ ; lim n→∞ τn = ∞ , lim n→∞ σn = ∞ ; σn ≤ n , σn is nondecreasing; ∆τn ≥ τ0 > 0 ; τσ = στ ; f (u) u ≥ m > 0 for u ̸ = 0. Moreover, we provide some examples to illustrate our main results.
In this paper, we study the statistical approximation properties of real and complex Post-Widder ... more In this paper, we study the statistical approximation properties of real and complex Post-Widder operators based on q-integers. We also obtain a Voronovskaya-type formula in statistical sense for these operators.
The aim of this work is to introduce split Pell and split Pell-Lucas quaternions. We give generat... more The aim of this work is to introduce split Pell and split Pell-Lucas quaternions. We give generating functions and Binet formulas for these numbers. Also, we obtain many identities for split Pell and split Pell-Lucas quaternions including Catalan's identity, Cassini's identity and d'Ocagne's identity.
Halici (Adv Appl Clifford Algebr 25(4):905–914, 2015) defined dual Fibonacci and dual Lucas octon... more Halici (Adv Appl Clifford Algebr 25(4):905–914, 2015) defined dual Fibonacci and dual Lucas octonions by the relations $${\widetilde{Q}_{n}={Q}_n+\varepsilon Q_{n+1}}$$Q~n=Qn+εQn+1 and $${\widetilde{P}_n=P_n+\varepsilon P_{n+1}}$$P~n=Pn+εPn+1 for every integer n where $${Q_n}$$Qn and $${P_n}$$Pn are the Fibonacci and Lucas octonions respectively, and $${\varepsilon}$$ε is the dual unit. The aim of this paper is to investigate properties of dual Fibonacci and dual Lucas octonions. After obtaining the Binet formulas for the sequences $${\{\widetilde{Q}_n \}_{n=0}^\infty}$${Q~n}n=0∞ and $${\{\widetilde{P}_n \}_{n=0}^\infty}$${P~n}n=0∞, we derive some identities for these sequences such as Catalan’s, Cassini’s and d’Ocagne’s identities.
In this article, we give the most general form of the quaternions algebra depending on 3-paramete... more In this article, we give the most general form of the quaternions algebra depending on 3-parameters. We define 3-parameter generalized quaternions (3PGQs) and study on various properties and applications. Firstly we present the definiton, the multiplication table and other properties of 3PGQs such as addition-substraction, multiplication and multiplication by scalar operations, unit and inverse elements, conjugate and norm. We give matrix representation and Hamilton operators for 3PGQs.We get polar represenation, De Moivre's and Euler's formulas with the matrix representations for 3PGQs. Besides, we give relations among the powers of the matrices associated with 3PGQs. Finally, Lie group and Lie algebra are studied and their matrix representations are shown. Also the Lie multiplication and the killing bilinear form are given.
Uploads
Papers by Zafer Ünal