scholarly journals On the connections between Pell numbers and Fibonacci p-numbers

2021 ◽  
Vol 27 (1) ◽  
pp. 148-160
Author(s):  
Anthony G. Shannon ◽  
◽  
Özgür Erdağ ◽  
Ömür Deveci ◽  
◽  
...  
Keyword(s):  
Pell Numbers ◽  
Binet Formula ◽  
Generating Matrix ◽  
Combinatorial Representation

In this paper, we define the Fibonacci–Pell p-sequence and then we discuss the connection of the Fibonacci–Pell p-sequence with the Pell and Fibonacci p-sequences. Also, we provide a new Binet formula and a new combinatorial representation of the Fibonacci–Pell p-numbers by the aid of the n-th power of the generating matrix of the Fibonacci–Pell p-sequence. Furthermore, we derive relationships between the Fibonacci–Pell p-numbers and their permanent, determinant and sums of certain matrices.

scholarly journals Matrix Manipulations for Properties of Pell p-Numbers and their Generalizations

2020 ◽  
Vol 28 (3) ◽  
pp. 89-102
Author(s):  
Özgür Erdağ ◽  
Ömür Deveci ◽  
Anthony G. Shannon
Keyword(s):  
Binet Formula ◽  
Generating Matrix ◽  
Combinatorial Representation

AbstractIn this paper, we define the Pell-Pell p-sequence and then we discuss the connection of the Pell-Pell p-sequence with Pell and Pell p-sequences. Also, we provide a new Binet formula and a new combinatorial representation of the Pell-Pell p-numbers by the aid of the nth power of the generating matrix the Pell-Pell p-sequence. Furthermore, we obtain an exponential representation of the Pell-Pell p-numbers and we develop relationships between the Pell-Pell p-numbers and their permanent, determinant and sums of certain matrices.

scholarly journals Summing a Family of Generalized Pell Numbers

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Linear Combination ◽  
Computer Algebra ◽  
Generating Functions ◽  
Fibonacci Numbers ◽  
Partial Sums ◽  
Pell Numbers ◽  
New Family ◽  
Binet Formula

AbstractA new family of generalized Pell numbers was recently introduced and studied by Bród ([2]). These numbers possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of generalized Pell numbers can be summed explicitly. For this, as a first step, a power P lnis expressed as a linear combination of Pmn. The summation of such expressions is then manageable using generating functions. Since the new family contains a parameter R = 2r, the relevant manipulations are quite involved, and computer algebra produced huge expressions that where not trivial to handle at times.

scholarly journals Perrin’s bivariate and complex polynomials

2021 ◽  
Vol 27 (2) ◽  
pp. 70-78
Author(s):  
Renata Passos Machado Vieira ◽  
Milena Carolina dos Santos Mangueira ◽  
Francisco Regis Vieira Alves ◽  
Paula Maria Machado Cruz Catarino
Keyword(s):  
Generating Function ◽  
Complex Polynomials ◽  
Bivariate Polynomials ◽  
Binet Formula ◽  
Generating Matrix ◽  
Definition Of

In this article, a study is carried out around the Perrin sequence, these numbers marked by their applicability and similarity with Padovan’s numbers. With that, we will present the recurrence for Perrin’s polynomials and also the definition of Perrin’s complex bivariate polynomials. From this, the recurrence of these numbers, their generating function, generating matrix and Binet formula are defined.

scholarly journals Bi-Periodic Pell Sequence

2020 ◽  
pp. 136-144
Author(s):  
S. Uygun ◽  
H. Karataş
Keyword(s):  
Generating Function ◽  
Convergence Properties ◽  
Pell Numbers ◽  
Binet Formula

In this study, we introduce a new generalization of the Pell numbers which is called bi-periodic Pell sequences. We then proceed to find the Binet formula as well as the generating function for this sequence. The well-known Cassini, Catalan and the D’ocagne’s identities as well as some related binomial summation and sum formulas are also given. The convergence properties of the consecutive terms of this sequence is also examined.

On circulant like matrices properties involving Horadam, Fibonacci, Jacobsthal and Pell numbers

2021 ◽  
Vol 617 ◽  
pp. 100-120
Author(s):  
Enide Andrade ◽  
Dante Carrasco-Olivera ◽  
Cristina Manzaneda
Keyword(s):  
Pell Numbers

scholarly journals Generating matrix of the bi-periodic Lucas numbers

2017 ◽  
Author(s):  
Arzu Coskun ◽  
Necati Taskara
Keyword(s):  
Lucas Numbers ◽  
Generating Matrix

scholarly journals Combinatorial Expansions in $K$-Theoretic Bases

10.37236/2320 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Jason Bandlow ◽  
Jennifer Morse
Keyword(s):  
Generating Functions ◽  
Symmetric Functions ◽  
Schur Functions ◽  
Geometric Information ◽  
Combinatorial Description ◽  
Littlewood Polynomials ◽  
Stanley Symmetric Functions ◽  
K Theory ◽  
Combinatorial Representation

We study the class $\mathcal C$ of symmetric functions whose coefficients in the Schur basis can be described by generating functions for sets of tableaux with fixed shape.  Included in this class are the Hall-Littlewood polynomials, $k$-Schur functions, and Stanley symmetric functions; functions whose Schur coefficients encode combinatorial, representation theoretic and geometric information. While Schur functions represent the cohomology of the Grassmannian variety of $GL_n$, Grothendieck functions $\{G_\lambda\}$ represent the $K$-theory of the same space.  In this paper, we give a combinatorial description of the coefficients when any element of $\mathcal C$ is expanded in the $G$-basis or the basis dual to $\{G_\lambda\}$.

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