Pell-Lucas Numbers Research Articles

On Mixed 𝐵-Concatenations of Pell and Pell–Lucas Numbers which are Pell Numbers

Let (𝑃𝑛)𝑛≥0 and (𝑄𝑛)𝑛≥0 be the Pell and Pell–Lucas sequences. Let 𝑏 be a positive integer such that 𝑏 ≥ 2. In this paper, we prove that the following two Diophantine equations 𝑃𝑛 = 𝑏𝑑𝑃𝑚 + 𝑄𝑘 and 𝑃𝑛 = 𝑏𝑑𝑄𝑚 + 𝑃𝑘 with 𝑑, the number of digits of 𝑃𝑘 or 𝑄𝑘 in base 𝑏, have only finitely many solutions in nonnegative integers (𝑚, 𝑛, 𝑘, 𝑏, 𝑑). Also, we explicitly determine these solutions in cases 2 ≤ 𝑏 ≤ 10.

Non-Newtonian Pell and Pell-Lucas numbers

In the present paper, we introduce a new type of Pell and Pell-Lucas numbers in terms of non-Newtonian calculus, which we call non-Newtonian Pell and non-Newtonian Pell-Lucas numbers, respectively. In non-Newtonian calculus, we study some significant identities and formulas for classical Pell and Pell-Lucas numbers. Therefore, we derive some relations with non-Newtonian Pell and Pell-Lucas numbers. Furthermore, we investigate some properties of non-Newtonian Pell and Pell-Lucas numbers, including Catalan-like identities, Cassini-like identities, Binet-like formulas, and generating functions.

Repdigits as Products of Consecutive Pell or Pell–Lucas Numbers

Repdigits as Products of Consecutive Pell or Pell–Lucas Numbers

On hyper-dual vectors and angles with Pell, Pell-Lucas numbers

<p>In this paper, we introduce two types of hyper-dual numbers with components including Pell and Pell-Lucas numbers. This novel approach facilitates our understanding of hyper-dual numbers and properties of Pell and Pell-Lucas numbers. We also investigate fundamental properties and identities associated with Pell and Pell-Lucas numbers, such as the Binet-like formulas, Vajda-like, Catalan-like, Cassini-like, and d'Ocagne-like identities. Furthermore, we also define hyper-dual vectors by using Pell and Pell-Lucas vectors and discuse hyper-dual angles. This extensionis not only dependent on our understanding of dual numbers, it also highlights the interconnectedness between integer sequences and geometric concepts.</p>

ON DUAL BICOMPLEX BALANCING AND LUCAS-BALANCING NUMBERS

In this paper, dual bicomplex Balancing and Lucas-Balancing numbers are defined, and some identities analogous to the classic properties of the Fibonacci and Lucas sequences are produced. We give the relationship between these numbers and Pell and Pell-Lucas numbers. From these, the basic bicomplex properties for the norm and its conjugate of these numbers are also developed. These in turn lead to the Binet formula, the generating functions and exponential generating functions, which are important concepts for number sequences. The Cassini identity, which is important for number sequences, actually emerged to solve the famous Curry paradox. We calculated the Cassini, Catalan, Vajda and d’Ocagne identities for these numbers.

On Some Properties of Bihyperbolic Numbers of The Lucas Type

To date, many authors in the literature have worked on special arrays in various computational systems. In this article, Lucas type bihyperbolic numbers were defined and their algebraic properties were examined. Bihyperbolic Lucas numbers were studied by Azak in 2021. Therefore, we only examined bihyperbolic Jacobsthal-Lucas and Pell-Lucas numbers. We also gave properties of bihyperbolic Jacobstal-Lucas and bihyperbolic Pell-Lucas numbers such as recursion relation, derivation function, Binet formula, D'Ocagne identity, Cassini identity and Catalan identity.

Pell and Pell–Lucas numbers as difference of two repdigits

Pell and Pell–Lucas numbers as difference of two repdigits

The Solution of a System of Higher-Order Difference Equations in Terms of Balancing Numbers

In this paper, we are interested in the closed-form solution of the following system of nonlinear difference equations of higher order,un+1 = 1/34-vn-m , vn+1 = 1/34-un-m, n, m ∈ N0,and the initial values u-j and v-j , j∈{0, 1, ..., m} are real numbers do not equal 34. We show that the solutions of this system are associated with Balancing numbers. As consequence, these solutions are also associated with Pell numbers, Pell-Lucas numbers, and Lucas-Balancing numbers. It is shown that the global stability of positive solutions of this system holds. Our results are illustrated via numerical examples.

On a new family of the generalized Gaussian k-Pell-Lucas numbers and their polynomials

In this paper, we generalize the known Gaussian Pell-Lucas numbers, and call such numbers as the generalized Gaussian k-Pell-Lucas numbers. We obtain relations between the family of the generalized Gaussian k-Pell-Lucas numbers and the known Gaussian Pell-Lucas numbers. We generalize the known Gaussian Pell-Lucas polynomials, and call such polynomials as the generalized Gaussian k-Pell-Lucas polynomials. We obtain relations between the family of the generalized Gaussian k-Pell-Lucas polynomials and the known Gaussian Pell-Lucas polynomials. In addition, we present the new generalizations of these numbers and polynomials in matrix form. Then, we get Cassini’s identities for these numbers and polynomials.

On applications of Pell and Pell-Lucas numbers with matrix method

In this study, new matrices which produce the Pell and Pell-Lucas numbers are given. By using these matrices, new identities and relations related to the Pell and Pell-Lucas numbers are obtained.

Some Matrix Applications on the Special Integer Number Sequences

In this paper, the matrices related to Fibonacci, Lucas, Pell, and Pell-Lucas numbers have been examined. By using these matrices new identities related to these integer sequences have been investigated.

On determinants, inverses, norms, and spread of skew circulant matrices involving the product of Pell and Pell-Lucas numbers

On determinants, inverses, norms, and spread of skew circulant matrices involving the product of Pell and Pell-Lucas numbers

Gaussian-bihyperbolic Numbers Containing Pell and Pell-Lucas Numbers

In this study, we define a new type of Pell and Pell-Lucas numbers which are called Gaussian-bihyperbolic Pell and Pell-Lucas numbers. We also define negaGaussian-bihyperbolic Pell and Pell-Lucas numbers. Moreover, we obtain Binet’s formulas, generating function formulas, d’Ocagne’s identities, Catalan’s identities, Cassini’s identities and some sum formulas for these new type numbers and we investigate some algebraic proporties of these. Furthermore, we give the matrix representation of Gaussian-bihyperbolic Pell and Pell-Lucas numbers.

Pell and Pell-Lucas hybrid quaternions

In this paper, we investigate Pell and Pell-Lucas numbers what new properties we can obtain by working on a new number system-hybrid quaternions. Firstly, we present some new identities of Pell and Pell-Lucas numbers and quaternions. Then, with the help of these identities and previously known identities, we get new identities on Pell and Pell-Lucas hybrid quaternions, including Binet?s and Cassini?s identities.

Dual-Gaussian Pell and Pell-Lucas numbers

In this study, we define a new type of Pell and Pell-Lucas numbers which are called dual-Gaussian Pell and dual-Gaussian Pell-Lucas numbers. We also give the relationship between negadual-Gaussian Pell and Pell-Lucas numbers and dual-complex Pell and Pell-Lucas numbers. Also, some sum ve product properties of Pell and Pell-Lucas numbers are given. Moreover, we obtain the Binet’s formula, generating function, d’Ocagne’s identity, Catalan’s identity, Cassini’s identity and some sum formulas for these new type numbers. Some algebraic proporties of dual-Gaussian Pell and Pell-Lucas numbers are investigated. Futhermore, we give the matrix representation of dual-Gaussian Pell and Pell-Lucas numbers.

Pell and Pell–Lucas Numbers as Product of Two Repdigits

Pell and Pell–Lucas Numbers as Product of Two Repdigits

On generalized (k, r)-Pell and (k, r)-Pell–Lucas numbers

We introduce new kinds of k-Pell and k-Pell–Lucas numbers related to the distance between numbers by a recurrence relation and show their relation to the (k,r)-Pell and (k,r)-Pell–Lucas numbers. These sequences differ both according to the value of the natural number k and the value of a new parameter r in the definition of this distance. We give several properties of these sequences. In addition, we establish the generating functions, some important identities, as well as the sum of the terms of the generalized (k,r)-Pell and (k,r)-Pell–Lucas numbers. Furthermore, we indicate another way to obtain the generalized (k,r)-Pell and (k,r)-Pell–Lucas sequences from the generating function, in connection to graphs.

Pell–Lucas Numbers as Sum of Same Power of Consecutive Pell Numbers

Pell–Lucas Numbers as Sum of Same Power of Consecutive Pell Numbers

On the problem of Pillai with Pell numbers, Pell–Lucas numbers and powers of 3

Let [Formula: see text] be the sequence of Pell numbers defined by [Formula: see text], [Formula: see text] and [Formula: see text] for all [Formula: see text] and let [Formula: see text] be its companion sequence, the Pell–Lucas numbers defined by [Formula: see text] and [Formula: see text] for all [Formula: see text]. In this paper, we find all integers [Formula: see text] admitting at least two representations as a difference between a Pell number or a Pell–Lucas number and a power of [Formula: see text]

On certain bihypernomials related to Pell and Pell-Lucas numbers

The bihyperbolic numbers are extension of hyperbolic numbers to four dimensions. In this paper we introduce the concept of Pell and Pell-Lucas bihypernomials as a generalization of bihyperbolic Pell and Pell-Lucas numbers, respectively.