scholarly journals On Generalized Jacobsthal and Jacobsthal-Lucas polynomials

2016 ◽  
Vol 24 (3) ◽  
pp. 61-78 ◽  
Author(s):  
Paula Catarino ◽  
Maria Luisa Morgado
Keyword(s):  
Generating Functions ◽  
Special Kind ◽  
Generalized Polynomials ◽  
Lucas Polynomials ◽  
Tridiagonal Matrices

AbstractIn this paper we introduce a generalized Jacobsthal and Jacobsthal-Lucas polynomials, Jh,nand jh,n, respectively, that consist on an extension of Jacobsthal's polynomials Jn(𝑥) and Jacobsthal-Lucas polynomials jn(𝑥). We provide their properties and a generalization of the usual identities. We also present, for each one of these generalized polynomials, their ordinary generating functions and matrices. In the last part of the paper, we present some special kind of tridiagonal matrices whose entries are elements of these generalized polynomials.

scholarly journals Some Properties of the(p,q)-Fibonacci and(p,q)-Lucas Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
GwangYeon Lee ◽  
Mustafa Asci
Keyword(s):  
Generating Functions ◽  
Combinatorial Identities ◽  
Fibonacci Polynomials ◽  
Lucas Polynomials ◽  
Pascal Matrix ◽  
Riordan Arrays ◽  
Combinatorial Sums

Riordan arrays are useful for solving the combinatorial sums by the help of generating functions. Many theorems can be easily proved by Riordan arrays. In this paper we consider the Pascal matrix and define a new generalization of Fibonacci polynomials called(p,q)-Fibonacci polynomials. We obtain combinatorial identities and by using Riordan method we get factorizations of Pascal matrix involving(p,q)-Fibonacci polynomials.

Fibonacci and Lucas polynomials

1981 ◽  
Vol 90 (3) ◽  
pp. 385-387 ◽  
Author(s):  
B. G. S. Doman ◽  
J. K. Williams
Keyword(s):  
Difference Equation ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Linear Difference Equation ◽  
Second Order ◽  
Lucas Numbers ◽  
Order Linear ◽  
Lucas Polynomials ◽  
Fibonacci And Lucas Numbers

The Fibonacci and Lucas polynomials Fn(z) and Ln(z) are denned. These reduce to the familiar Fibonacci and Lucas numbers when z = 1. The polynomials are shown to satisfy a second order linear difference equation. Generating functions are derived, and also various simple identities, and relations with hypergeometric functions, Gegenbauer and Chebyshev polynomials.

scholarly journals Symmetric and generating functions of generalized (p,q)-numbers

2021 ◽  
Vol 48 (4) ◽  
Author(s):  
Nabiha Saba ◽  
◽  
Ali Boussayoud ◽  
Abdelhamid Abderrezzak ◽  
◽  
...  
Keyword(s):  
Generating Functions ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Pell Numbers ◽  
Lucas Polynomials ◽  
Binet’S Formula

In this paper, we will firstly define a new generalization of numbers (p, q) and then derive the appropriate Binet's formula and generating functions concerning (p,q)-Fibonacci numbers, (p,q)- Lucas numbers, (p,q)-Pell numbers, (p,q)-Pell Lucas numbers, (p,q)-Jacobsthal numbers and (p,q)-Jacobsthal Lucas numbers. Also, some useful generating functions are provided for the products of (p,q)-numbers with bivariate complex Fibonacci and Lucas polynomials.

scholarly journals On some properties of generalized Fibonacci and Lucas polynomials

2017 ◽  
Vol 7 (2) ◽  
pp. 216-224
Author(s):  
Sümeyra Uçar
Keyword(s):  
Lucas Polynomials ◽  
Tridiagonal Matrices ◽  
Laplace Expansion

In this paper we investigate some properties of generalized Fibonacci and Lucas polynomials. We give some new identities using matrices and Laplace expansion for the generalized Fibonacci and Lucas polynomials. Also, we introduce new families of tridiagonal matrices whose successive determinants generate any subsequence of these polynomials.

scholarly journals On the (p,q)–Chebyshev Polynomials and Related Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 136 ◽  
Author(s):  
Can Kızılateş ◽  
Naim Tuğlu ◽  
Bayram Çekim
Keyword(s):  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Lucas Polynomials

In this paper, we introduce ( p , q ) –Chebyshev polynomials of the first and second kind that reduces the ( p , q ) –Fibonacci and the ( p , q ) –Lucas polynomials. These polynomials have explicit forms and generating functions are given. Then, derivative properties between these first and second kind polynomials, determinant representations, multilateral and multilinear generating functions are derived.

scholarly journals Symmetric and generating functions for some generalized polynomials

2020 ◽  
Vol 8 (4) ◽  
pp. 1756-1765
Author(s):  
Saba N. ◽  
Boussayoud A. ◽  
Chelgham M.
Keyword(s):  
Generating Functions ◽  
Generalized Polynomials

scholarly journals New Families of Three-Variable Polynomials Coupled with Well-Known Polynomials and Numbers

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 264 ◽  
Author(s):  
Can Kızılateş ◽  
Bayram Çekim ◽  
Naim Tuğlu ◽  
Taekyun Kim
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Generating Functions ◽  
Explicit Representation ◽  
Partial Differential ◽  
Generalized Polynomials ◽  
Special Cases

In this paper, firstly the definitions of the families of three-variable polynomials with the new generalized polynomials related to the generating functions of the famous polynomials and numbers in literature are given. Then, the explicit representation and partial differential equations for new polynomials are derived. The special cases of our polynomials are given in tables. In the last section, the interesting applications of these polynomials are found.

scholarly journals A NEW FAMILIES OF GAUSS k-JACOBSTHAL NUMBERS AND GAUSS k-JACOBSTHAL-LUCAS NUMBERS AND THEIR POLYNOMIALS

2020 ◽  
Vol 20 (4) ◽  
pp. 893-908
Author(s):  
ENGİN ÖZKAN ◽  
MERVE TAŞTAN
Keyword(s):  
Matrix Representation ◽  
Lucas Numbers ◽  
Generalized Polynomials ◽  
Lucas Polynomials ◽  
The Family

In this paper, we define the new families of Gauss k-Jacobsthal numbers and Gauss k-Jacobsthal-Lucas numbers. We obtain some exciting properties of the families. We give the relationships between the family of the Gauss k-Jacobsthal numbers and the known Gauss Jacobsthal numbers, the family of the Gauss k-Jacobsthal-Lucas numbers and the known Gauss Jacobsthal-Lucas numbers. We also define the generalized polynomials for these numbers. Further, we obtain some interesting properties of the polynomials. In addition, we give the relationships between the generalized Gauss k-Jacobsthal polynomials and the known Gauss Jacobsthal polynomials, the generalized Gauss k-Jacobsthal-Lucas polynomials and the known Gauss Jacobsthal-Lucas polynomials. Furthermore, we find the new generalizations of these families and the polynomials in matrix representation. Then we prove Cassini’s Identities for the families and their polynomials.

scholarly journals Parity Theorems for Statistics on Domino Arrangements

10.37236/1977 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Mark A. Shattuck ◽  
Carl G. Wagner
Keyword(s):  
Generating Functions ◽  
Lucas Polynomials ◽  
Special Values

We study special values of Carlitz's $q$-Fibonacci and $q$-Lucas polynomials $F_n(q,t)$ and $L_n(q,t)$. Brief algebraic and detailed combinatorial treatments are presented, the latter based on the fact that these polynomials are bivariate generating functions for a pair of statistics defined, respectively, on linear and circular domino arrangements.

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