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.
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 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.
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.
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.