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Electronic Journal of Mathematical Analysis and Applications2
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Journal of Contemporary Applied Mathematics2
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ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA1
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BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE1
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CREAT. MATH. INFORM.1
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Complex Analysis and its Synergies1
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Demonstratio Mathematica1
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Jordan Journal of Mathematics and Statistics1
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Journal of New Theory1
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Journal of Universal Mathematics1
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Electronic Journal of Mathematical Analysis and Applications2
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Journal of Contemporary Applied Mathematics2
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Springer2
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ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA1
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Aligarh Muslim University1
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BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE1
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CREAT. MATH. INFORM.1
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DE GRUYTER1
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Journal of New Theory1
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Journal of Universal Mathematics1
On the Hyperbolic Leonardo and Hyperbolic Francois Quaternions
Dişkaya, Orhan | Menken, Hamza
In this paper, we present a new definition, referred to as the Francois sequence, related to the Lucas-like form of the Leonardo sequence. We also introduce the hyperbolic Leonardo and hyperbolic Francois quaternions. Afterward, we derive the Binet-like formulas and their generating functions. Moreover, we provide some binomial sums, Honsberger-like, d’Ocagne-like, Catalan-like, and Cassini-like identities of the hyperbolic Leonardo quaternions and hyperbolic Francois quaternions that allow an understanding of the quaternions’ proper- ties and their relation to the Francois sequence and Leonardo sequence. Finally, considering the results presented in this study, we discuss the need for further research in this field.
On the Quinary Fibonacci-Padovan Sequences
Dişkaya, Orhan | Menken, Hamza
In this paper, we consider the Fibonacci and Padovan sequences. We introduce the quinary Fibonacci-Padovan sequences whose compounds are the Fibonacci and Padovan sequences. We derive the Binet-like formulas, the generating functions and exponential generating functions of these sequences. Also, we obtain some binomial identities, series and sums for them.
On the Fibonacci quaternion sequence with quadruple-produce components
Dişkaya, Orhan | Menken, Hamza
This paper examines the Fibonacci quaternion sequence with quadruple-produce components, and demonstrates a golden-like ra- tio and some identities for this sequence. Its generating and exponential generating functions are given. Along with these, its series and binomial sum formula are established.
On the Quadra Fibona-Pell and Hexa Fibona-Pell-Jacobsthal Sequences
Dişkaya, Orhan | Menken, Hamza
In this paper, we consider the Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas sequences. We introduce the quadra Fibona-Pell,Fibona-Jacobsthal and Pell-Jacobsthal and the hexa Fibona-Pell- Jacobsthal sequences whose compounds are the Fibonacci, Pell and Jacobsthal sequences. We derive the Binet-like formulas, the generating functions and the exponential generating functions of these sequences. Also, we obtain some binomial identities for them.
On the incomplete narayana numbers
Dişkaya, Orhan | Menken, Hamza
In this paper, we first express with sums of binomial coefficients of the Narayana sequence. Moreover, we define the incomplete Narayana numbers and examine their recurrence relations, some properties of these numbers, and the generating function of the incomplete Narayana numbers.
ARCTANGENT IDENTITIES INVOLVING THE JACOBSTHAL AND JACOBSTHAL-LUCAS NUMBERS
Dişkaya, Orhan
This study presents novel arctangent identities that establish connections between the Jacobsthal and Jacobsthal-Lucas numbers. These findings contribute to the understanding of the interplay between trigonometric functions and number theory, particularly in relation to well-known mathematical sequences and constants.
ON THE RECURRENCES OF THE JACOBSTHAL SEQUENCE
Dişkaya, Orhan | Menken, Hamza
In the present work, two new recurrences of the Jacobsthal sequence are defined. Some identities of these sequences which we call the Jacobsthal array is examined. Also, the generating and series functions of the Jacobsthal array are obtained.
Degenerate Bernoulli–Fibonacci and Euler–Fibonacci polynomials
Dişkaya, Orhan
In this article, we first introduce the degenerate Bernoulli–Fibonacci numbers and degenerate Euler–Fibonacci numbers. Using these definitions, we then define the degenerate Bernoulli–Fibonacci polynomials and degenerate Euler–Fibonacci polynomials and examine their graphs for several initial values of λ. Subsequently, we define the degenerate Bernoulli and Euler F-polynomials and derive new exponential generating functions for these polynomials. Additionally, we investigate various identities associated with these polynomials.
Fibonacci-based generalizations of degenerate Stirling numbers
Dişkaya, Orhan
This paper introduces and systematically investigates Fibonacci-based analogues and generalizations of degenerate Stirling and Lah numbers. We begin by recalling the classical definitions and key properties of Stirling numbers of both kinds, Lah numbers, Fibonacci numbers, and Fibonomial coefficients, along with F-falling and F-rising factorials. The foundational concept of degenerate numbers and their associated degenerate factorials, as initiated by Carlitz, is also reviewed. Our primary contribution is the definition of four new families of numbers: the degenerate F-Stirling numbers of the first kind and S1,λF(n,k), the degenerate F-Stirling numbers of the second kind S2,λF(n,k), and the degenerate F-Lah number LλF(n,k). These numbers are precisely characterized as connection coeffici...
ON THE (p, q)− FIBONACCI N−DIMENSIONAL RECURRENCES
Dişkaya, Orhan | Menken, Hamza
In this study, one-dimensional, two-dimensional, three-dimensional and n−dimensional recurrences of the (p, q)−Fibonacci sequence are examined and their some identities are given.
On the Pseudo-Fibonacci and Pseudo-Lucas Quaternions
Dişkaya, Orhan | Menken, Hamza
There are a lot of quaternion numbers that are related to theFibonacci and Lucas numbers or their generalizations have been described and extensively explored. The coefficients of these quaternions have been chosen from terms of Fibonacci and Lucas numbers. In this study, we define two new quaternions that are pseudo-Fibonacci and pseudo-Lucas quaternions. Then, we give their Binet-like formulas, generating functions, certain binomial sums and Honsberg-like, d’Ocagne-like, Catalan-like and Cassini-like identities.
On the Incomplete (p,q)−Fibonacci and (p,q)−Lucas Numbers
Dişkaya, Orhan | Menken, Hamza
In this present work, the incomplete (p,q)−Fibonacci and (p,q)−Lucas numbers are defined. We examine their recurrence relations as well as some of their properties. We derive their generating functions.
Compositions of positive integers with 2s and 3s
Dişkaya, Orhan | Menken, Hamza
In this article, we consider compositions of positive integers with 2s and 3s. We see that these compositions lead us to results that involve Padovan numbers, and we give some tiling models of these composi- tions. Moreover, we examine some tiling models of the compositions related to the Padovan polynomials and prove some identities using the tiling model’s method. Next, we obtain various identities of the compositions of positive integers with 2s and 3s related to the Padovan numbers. The number of palindromic compositions of this type is determined, and some numerical arithmetic functions are defined. Finally, we provide a table that compares all of the results obtained from compositions of positive integers with 2s and 3s.
On the Non-Newtonian Padovan and Non-Newtonian Perrin Numbers
Dişkaya, Orhan
In this work, we introduce a novel version of Padovan and Perrin numbers which we refer to as non-Newtonian Padovan and non-Newtonian Perrin numbers. Furthermore, we examine about a number of their properties. Additionally, we provide a variety of identities and formulas involving these new kinds, including the Binet-like formulas, the generating functions, the partial sum formulas, and the binomial sum formulas.
On the Lagrange Interpolations of the Jacobsthal and Jacobsthal-Lucas Sequences
Dişkaya, Orhan
This study explores the formation of polynomials of at most degree nusing the first n+1 terms of the Jacobsthal and Jacobsthal-Lucas sequences through Lagrange interpolation. The paper provides a detailed examination of the recurrence relations and various identities associated with the Jacobsthal and Jacobsthal-Lucas Lagrange Interpolation Polynomials.
THE CLASSICAL AES-LIKE CRYPTOLOGY VIA THE FIBONACCI POLYNOMIAL MATRIX
Dişkaya, Orhan | Menken, Hamza
Galois field, has an important position in cryptology. Advanced Encryption Standard (AES) also used in polynomial operations. In this paper, we consider the polynomial operations on the Galois fields, the Fibonacci polynomial sequences. Using a certain irreducible polynomial, we redefine the elements of Fibonacci polynomial sequences to use in our cryptology algorithm. So, we find the classical AES-like cryptology via the Fibonacci polynomial matrix. Successful results were achieved with the method used.