scholarly journals The generalized k-Fibonacci polynomials and generalized k-Lucas polynomials

2021 ◽  
Vol 27 (2) ◽  
pp. 148-158
Author(s):  
Merve Taştan ◽  
◽  
Engin Özkan ◽  
Anthony G. Shannon ◽  
◽  
...  
Keyword(s):  
Matrix Representation ◽  
Fibonacci Polynomials ◽  
Lucas Polynomials ◽  
The Family ◽  
Generalized Lucas Polynomials

In this paper, we define new families of Generalized Fibonacci polynomials and Generalized Lucas polynomials and develop some elegant properties of these families. We also find the relationships between the family of the generalized k-Fibonacci polynomials and the known generalized Fibonacci polynomials. Furthermore, we find new generalizations of these families and the polynomials in matrix representation. Then we establish Cassini’s Identities for the families and their polynomials. Finally, we suggest avenues for further research.

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.

On a new family of gauss k-Lucas numbers and their polynomials

2020 ◽  
pp. 2150101
Author(s):  
Engi̇n Özkan ◽  
Merve Taştan
Keyword(s):  
Matrix Representation ◽  
Lucas Numbers ◽  
Generalized Polynomials ◽  
Lucas Polynomials ◽  
New Family ◽  
The Family

In this paper, we define a new family of Gauss [Formula: see text]-Lucas numbers. We give the relationships between the family of the Gauss [Formula: see text]-Lucas numbers and the known Gauss Lucas numbers. We also define the generalized polynomials for these numbers. We obtain some interesting properties of the polynomials. We also give the relationships between the generalized Gauss [Formula: see text]-Lucas polynomials and the known Gauss Lucas polynomials. Furthermore, we find 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 Distance Fibonacci Polynomials—Part II

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1723
Author(s):  
Urszula Bednarz ◽  
Małgorzata Wołowiec-Musiał
Keyword(s):  
Generating Function ◽  
Initial Conditions ◽  
Symmetric Matrix ◽  
Fibonacci Polynomials ◽  
Lucas Polynomials ◽  
Graph Interpretation ◽  
Q Matrix ◽  
Generalized Lucas Polynomials ◽  
Symmetric Patterns

In this paper we use a graph interpretation of distance Fibonacci polynomials to get a new generalization of Lucas polynomials in the distance sense. We give a direct formula, a generating function and we prove some identities for generalized Lucas polynomials. We present Pascal-like triangles with left-justified rows filled with coefficients of these polynomials, in which one can observe some symmetric patterns. Using a general Q-matrix and a symmetric matrix of initial conditions we also define matrix generators for generalized Lucas 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.

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