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 Distance Fibonacci Polynomials by Graph Methods

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2075
Author(s):  
Dominik Strzałka ◽  
Sławomir Wolski ◽  
Andrzej Włoch
Keyword(s):  
Initial Conditions ◽  
Symmetric Matrix ◽  
Fibonacci Polynomials ◽  
Pascal’S Triangle ◽  
Graph Interpretation ◽  
Symmetric Structure ◽  
Binomial Formula ◽  
Narayana Numbers ◽  
Pascal's Triangle

In this paper we introduce and study a new generalization of Fibonacci polynomials which generalize Fibonacci, Jacobsthal and Narayana numbers, simultaneously. We give a graph interpretation of these polynomials and we obtain a binomial formula for them. Moreover by modification of Pascal’s triangle, which has a symmetric structure, we obtain matrices generated by coefficients of generalized Fibonacci polynomials. As a consequence, the direct formula for generalized Fibonacci polynomials was given. In addition, we determine matrix generators for generalized Fibonacci polynomials, using the symmetric matrix of initial conditions.

scholarly journals Distance Fibonacci Polynomials

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1540
Author(s):  
Urszula Bednarz ◽  
Małgorzata Wołowiec-Musiał
Keyword(s):  
Generating Function ◽  
Fibonacci Polynomials ◽  
Pascal’S Triangle ◽  
Graph Interpretation ◽  
Pascal's Triangle

In this paper, we introduce a new kind of generalized Fibonacci polynomials in the distance sense. We give a direct formula, a generating function and matrix generators for these polynomials. Moreover, we present a graph interpretation of these polynomials, their connections with Pascal’s triangle and we prove some identities for them.

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 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.

scholarly journals Incomplete Bivariate Fibonacci and Lucas -Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Dursun Tasci ◽  
Mirac Cetin Firengiz ◽  
Naim Tuglu
Keyword(s):  
Generating Function ◽  
Lucas Numbers ◽  
Lucas Polynomials ◽  
Fibonacci And Lucas Numbers

We define the incomplete bivariate Fibonacci and Lucas polynomials. In the case , , we obtain the incomplete Fibonacci and Lucas numbers. If , , we have the incomplete Pell and Pell-Lucas numbers. On choosing , , we get the incomplete generalized Jacobsthal number and besides for the incomplete generalized Jacobsthal-Lucas numbers. In the case , , , we have the incomplete Fibonacci and Lucas numbers. If , , , , we obtain the Fibonacci and Lucas numbers. Also generating function and properties of the incomplete bivariate Fibonacci and Lucas polynomials are given.

ON THE POSITIVENESS OF A FUNCTIONAL SYMMETRIC MATRIX USED IN DIGITAL FILTER DESIGN

2004 ◽  
Vol 13 (05) ◽  
pp. 1105-1110 ◽  
Author(s):  
YAN WU
Keyword(s):  
Signal Processing ◽  
Generating Function ◽  
Digital Filter ◽  
Simple Proof ◽  
Filter Design ◽  
Symmetric Matrix ◽  
Toeplitz Matrices ◽  
Positive Definite ◽  
Real Numbers ◽  
Digital Filter Design

This paper gives a simple proof for the positiveness of two important symmetric Toeplitz matrices used in communication and signal processing. It utilizes the shifting property of a so-called Uniformly Band-Restricted (UBR) function, which is the generating function for a generic functional symmetric matrix. It is shown that the functional symmetric matrix is positive definite if the UBR function is evaluated at a sequence of distinct real numbers.

Sign in / Sign up

Export Citation Format

Share Document