Binet Formulas Research Articles

Hyperbolic (s,t)-Fibonacci and (s,t)-Lucas Quaternions

In this study, we define hyperbolic (s,t)-Fibonacci and (s,t)-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 (s,t)-Fibonacci and (s,t)-Lucas quaternions. Finally, the terms of the (s,t)-Fibonacci and (s,t)-Lucas sequences are associated with their hyperbolic quaternion values.

A Study on Matrix Sequence with Generalized Guglielmo Numbers Components

We know that the diverse applications of matrix sequences in fields such as physics, engineering, architecture, nature, and art. Numerous authors have delved into the study of these matrix sequences in existing literature. In this study, we define and investigate the generalized Guglielmo matrix sequence. For this aim we explore four specific cases of that sequence that are called triangular matrix sequences, Lucas-triangular matrix sequences, oblong matrix sequences, and pentagonal matrix sequences. Next, we present Binet's formulas, generating functions, the summation formulas and some elementary identities for these sequences. Moreover, we give some identities and matrices related with these sequences. Furthermore, we show that there always exist some relationship between generalized triangular, Lucas-triangular, oblong and pentagonal matrix sequences.

A New Approach to k-Oresme and k-Oresme-Lucas Sequences

In this study, the k-Oresme and k-Oresme-Lucas sequences are defined, and some terms of these sequence are given. Then, the relations between the terms of the k-Oresme and k-Oresme-Lucas sequences are presented. In addition, we give these sequences the Binet formulas, generating functions, Cassini identity, Catalan identity etc. Moreover, the k-Oresme and k-Oresme-Lucas sequences are associated with Fibonacci, Pell numbers and Lucas, and Pell- Lucas numbers, respectively. Finally, the Catalan transforms of these sequences are given and Hankel transforms are applied to these Catalan sequences and associated with the terms of the sequence.

The Properties of Binomial Transforms for Modified (s,t)-Pell Matrix Sequence

In this study we investigate a generalization of modified Pell sequence, which is called (s,t)-modified Pell sequence. By considering this sequence, we define the matrix sequence whose elements are (s,t)-modified Pell numbers. Furthermore, we define various binomial transforms for modified (s,t)-Pell matrix sequence. Finally, we give some relationships for (s,t)-modified Pell matrix sequences such as Binet formulas, the generating functions and some sum formulas...

Binomial Transforms of the Third-Order Jacobsthal and Modified Third-Order Jacobsthal Polynomials

In this study, we define the binomial transforms of third-order Jacobsthal and modified third-order Jacobsthal polynomials. Further, the generating functions, Binet formulas and summation of these binomial transforms are found by recurrence relations. Also, we establish the relations between these transforms by deriving new formulas. Finally, the Vajda, d'Ocagne, Catalan and Cassini formulas for these transforms are obtained.

ON THE GENERALIZED DUAL FIBONACCI AND LUCAS OCTONIONS

Generalized number systems, particularly octonions with their algebraic structure, have drawn much interest in mathematics, physics, and computer technology. Therefore, in this paper, we introduce two new concepts, modified generalized dual Fibonacci and modified generalized dual Lucas octonions, to expand the topic of octonions. Additionally, we explore the well-known Catalan and Cassini identities, shedding light on the characteristics of these new constructs. Also, we give generating functions and the Binet formulas of the modified generalized dual Fibonacci and modified generalized dual Lucas octonions.

Copper ratio obtained by generalizing the Fibonacci sequence

In this study, we define a new generalization of the Fibonacci sequence that gives the copper ratio, and we will call it the copper Fibonacci sequence. In addition, inspired by the copper Fibonacci definition, we also define copper Lucas sequences, and then we give the relationships between the terms of these sequences. We present some properties, such as the Binet formulas, special summation formulas, special generating functions, etc. We find the relationships between the roots of the characteristic equation of these sequences and the general terms of these sequences. What is interesting here is that the relationships obtained from that between the roots of the characteristic equation of these new sequences and the terms of the sequences are satisfied in both roots. In addition, we examine the relationships between these sequences with the classic Fibonacci and Lucas sequences. Moreover, we calculate some identities of these sequences, such as Cassini and Catalan. Then Catalan transformation is applied to these sequences, and their terms are found. Furthermore, we apply Hankel transform to the Catalan transform of these sequences. Besides, we associate the terms of the Hankel transformation of the Catalan copper Fibonacci sequence with the classical Fibonacci numbers and the terms of the Hankel transformation of the Catalan copper Lucas sequence with the terms of the copper Lucas sequence. We present the application of copper Fibonacci and copper Lucas sequences to hyperbolic quaternions. Finally, the terms of the copper Fibonacci and copper Lucas sequences are associated with their hyperbolic quaternion values.

ON THE QUADRA MERSENNE-JACOBSTHAL, QUADRA MERSENNE-PELL AND RELATED QUATERNION SEQUENCES

The aim of this paper is primarily to introduce the quadra Mersenne-Jacobsthal and quadra Mersenne-Pell sequences. Then, the Binet formulas and generating functions for these sequences are obtained. Moreover, quadra Mersenne-Jacobsthal and quadra Mersenne-Pell quaternions were defined and Binet formulas, generating functions, some quaternion sums and relations were obtained for these quaternions.

Generalization of the k-Leonardo sequence and their hyperbolic quaternions

In this study, we define the k-Leonardo, k-Leonardo-Lucas, and Modified k-Leonardo sequences, and some terms of these sequences are given. Then, we obtain the generating functions, summation formulas, etc. Also, we obtain the Binet formulas in three different ways. The first is in the known classical way, the second is with the help of the sequence's generating functions, and the third is with the help of the matrices. In addition, we examine the relations between the terms of the k-Leonardo, k-Leonardo-Lucas, Modified k-Leonardo, Leonardo, Leonardo-Lucas, Modified Leonardo, Francois, Fibonacci, and Lucas sequences. Moreover, we associate the terms of these sequences with matrices. Furthermore, we present on the application of these sequences to hyperbolic quaternions. For these quaternions, we give many properties such as Binet formulas. Finally, the terms of the k-Leonardo, k-Leonardo-Lucas, and Modified k-Leonardo sequences are associated with their hyperbolic quaternion values.

Some geometric properties of the Padovan vectors in Euclidean 3-space

Padovan numbers were defined by Stewart (1996) in honor of the modern architect Richard Padovan (1935) and were first discovered in 1924 by Gerard Cordonnier. Padovan numbers are a special status of Tribonacci numbers with initial conditions and general terms. The ratio between Padovan numbers is one of the important algebraic numbers because it produces plastic numbers. Up to now, various studies have been conducted on Padovan numbers and Padovan polynomial sequences. In this study, Padovan vectors are defined for the first time by using the Padovan Binet-like formula and reduction relation. Then, geometric properties of Padovan vectors such as inner product, norm, and vector products are analyzed. In the last part of the study, Padovan vectors were calculated with Binet formulas in the Geogebra program. In addition, the first ten Padovan numbers and Padovan vectors were calculated using the Binet formulas and shown as points and vectors in three-dimensional space. According to the Padovan vectors found, the Padovan curve was drawn in space for the first time by using the curve fitting feature of the Geogebra program. Thus, with our study, a geometric approach to Padovan number sequences was brought for the first time.

On Hybrid Hyper k-Pell, k-Pell–Lucas, and Modified k-Pell Numbers

Many different number systems have been the topic of research. One of the recently studied number systems is that of hybrid numbers, which are generalizations of other number systems. In this work, we introduce and study the hybrid hyper k-Pell, hybrid hyper k-Pell–Lucas, and hybrid hyper Modified k-Pell numbers. In order to study these new sequences, we established new properties, generating functions, and the Binet formula of the hyper k-Pell, hyper k-Pell–Lucas, and hyper Modified k-Pell sequences. Thus, we present some algebraic properties, recurrence relations, generating functions, the Binet formulas, and some identities for the hybrid hyper k-Pell, hybrid hyper k-Pell–Lucas, and hybrid hyper Modified k-Pell numbers.

On Some Combinatorial Properties of Oresme Hybrationals

In this paper, we study the Oresme hybrationals that generalize Oresme hybrid numbers and Oresme rational functions. We give a reccurence relation and a generating function for Oresme hybrationals. Moreover, we give some of their properties, among others, Binet formulas and general bilinear index-reduction formulas, through which we can obtain Catalan-, Cassini-, Vajda-, and d’Ocagne-type identities.

Hybrid Numbers with Fibonacci and Lucas Hybrid Number Coefficients

In this paper, we introduce hybrid numbers with Fibonacci and Lucas hybrid number coefficients. We give the Binet formulas, generating functions, and exponential generating functions for these numbers. Then we define an associate matrix for these numbers. In addition, using this matrix, we present two different versions of Cassini identity of these numbers.

On the Mersenne and Mersenne-Lucas hybrinomial quaternions

In this paper, we introduce Mersenne and Mersenne-Lucas hybrinomial quaternions and present some of their properties. Some identities are derived for these polynomials. Furthermore, we give the Binet formulas, Catalan, Cassini, d’Ocagne identity and generating and exponential generating function of these hybrinomial quaternions.

Construction of general forms of ordinary generating functions for more families of numbers and multiple variables polynomials

The aim of this paper is to construct general forms of ordinary generating functions for special numbers and polynomials involving Fibonacci type numbers and polynomials, Lucas numbers and polynomials, Chebyshev polynomials, Sextet polynomials, Humbert-type numbers and polynomials, chain and anti-chain polynomials, rank polynomials of the lattices, length of any alphabet of words, partitions, and other graph polynomials. By applying the Euler transform and the Lambert series to these generating functions, many new identities and relations are derived. By using differential equations of these generating functions, some new recurrence relations for these polynomials are found. Moreover, general Binet's type formulas for these polynomials are given. Finally, some new classes of polynomials and their corresponding certain family of special numbers are investigated with the help of these generating functions.

Special transforms of the generalized bivariate Fibonacci and Lucas polynomials

This paper deals with the Catalan, Hankel, binomial transforms of the generalized bivariate Fibonacci and Lucas polynomials. Also, some useful results such as generating functions, Binet formulas, summations of transforms defined by using recurrence relations of these special polynomials are presented. Furthermore, certain important relations among these transforms are deduced by using obtained new formulas. Finally, the Catalan, Cassini, Vajda and d'Ocagne formulas for these transforms are also derived.

Generalized Tribonacci Polynomials

In this paper, we investigate the generalized Tribonacci polynomials and we deal with, in detail, two special cases which we call them (r,s,t)-Tribonacci and (r,s,t)-Tribonacci-Lucas polynomials. We also introduce and investigate a new sequence and its two special cases namely the generalized co-Tribonacci, (r,s,t)-co-Tribonacci and (r,s,t)-co-Tribonacci-Lucas polynomials, respectively. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these polynomial sequences. Moreover, we give some identities and matrices related to these polynomials. Furthermore, we evaluate the infinite sums of special cases of (r,s,t)-Tribonacci and (r,s,t)-Tribonacci-Lucas polynomials.

On the Split Mersenne and Mersenne-Lucas Hybrid Quaternions

In this communication, introduce the split Mersenne and Mersenne-Lucas hybrid quaternions, also obtaining generating functions and Binet formulas for these hybrid quaternions and investigating some properties among them.

Bi-Periodic (p,q)-Fibonacci and Bi-Periodic (p,q)-Lucas Sequences

In this paper, we define bi-periodic (p,q)-Fibonacci and bi-periodic (p,q)-Lucas sequences, which generalize Fibonacci type, Lucas type, bi-periodic Fibonacci type and bi-periodic Lucas type sequences, using recurrence relations of (p,q)-Fibonacci and (p,q)-Lucas sequences. Generating functions and Binet formulas that allow us to calculate the nth terms of these sequences are given and the convergence properties of their consecutive terms are examined. Also, we prove some fundamental identities of bi-periodic (p,q)-Fibonacci and bi-periodic (p,q)-Lucas sequences conform to the well-known properties of Fibonacci and Lucas sequences.

Generalized Olivier Numbers

In this paper, we introduce and investigate the generalized Olivier sequences and we deal with, in detail, two special cases, namely, Olivier and Olivier-Lucas sequences. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these sequences. We also provide variousmatrices and identities associated with these sequences. Furthermore, we show that there are close relations between Olivier, Olivier-Lucas and adjusted Pell-Padovan, third order Lucas-Pell, third order Fibonacci-Pell, Pell-Perrin, Pell-Padovan numbers. Moreover, we give some identities and matrices related with thesesequences.