scholarly journals Closed Expressions of the Fibonacci Polynomials in Terms of Tridiagonal Determinants

2017 ◽  
Author(s):  
Feng Qi ◽  
Jing-Lin Wang ◽  
Bai-Ni Guo
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Polynomials ◽  
Closed Expression

In the paper, the authors nd a new closed expression for the Fibonacci polynomials and, consequently, for the Fibonacci numbers, in terms of a tridiagonal determinant.

scholarly journals Some Identities Involving Fibonacci Polynomials and Fibonacci Numbers

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 334 ◽  
Author(s):  
Yuankui Ma ◽  
Wenpeng Zhang
Keyword(s):  
Power Series ◽  
Structural Properties ◽  
Fibonacci Numbers ◽  
Second Order ◽  
Fibonacci Polynomials ◽  
Recursive Sequence ◽  
Combinatorial Methods

The aim of this paper is to research the structural properties of the Fibonacci polynomials and Fibonacci numbers and obtain some identities. To achieve this purpose, we first introduce a new second-order nonlinear recursive sequence. Then, we obtain our main results by using this new sequence, the properties of the power series, and the combinatorial methods.

scholarly journals Some Algebraic Aspects of Morse Code Sequences

2003 ◽  
Vol Vol. 6 no. 1 ◽  
Author(s):  
Johann Cigler
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Polynomials ◽  
Morse Code ◽  
Combinatorial Interpretations ◽  
International Audience

International audience Morse code sequences are very useful to give combinatorial interpretations of various properties of Fibonacci numbers. In this note we study some algebraic and combinatorial aspects of Morse code sequences and obtain several q-analogues of Fibonacci numbers and Fibonacci polynomials and their generalizations.

A multi-parameter generalization of the symmetric algorithm

2018 ◽  
Vol 68 (4) ◽  
pp. 699-712
Author(s):  
José L. Ramírez ◽  
Mark Shattuck
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Fibonacci Numbers ◽  
Recursive Formula ◽  
Binomial Coefficient ◽  
Fibonacci Polynomials ◽  
Combinatorial Argument ◽  
Recurrent Sequences ◽  
Seidel Method ◽  
Multivariate Generalization

Abstract The symmetric algorithm is a variant of the well-known Euler-Seidel method which has proven useful in the study of linearly recurrent sequences. In this paper, we introduce a multivariate generalization of the symmetric algorithm which reduces to it when all parameters are unity. We derive a general explicit formula via a combinatorial argument and also an expression for the row generating function. Several applications of our algorithm to the q-Fibonacci and q-hyper-Fibonacci numbers are discussed. Among our results is an apparently new recursive formula for the Carlitz Fibonacci polynomials. Finally, a (p, q)-analogue of the algorithm is introduced and an explicit formula for it in terms of the (p, q)-binomial coefficient is found.

scholarly journals Construction of a new class of generating functions of binary products of some special numbers and polynomials

Filomat ◽  
2021 ◽  
Vol 35 (3) ◽  
pp. 1001-1013
Author(s):  
Souhila Boughaba ◽  
Ali Boussayoud ◽  
Serkan Araci ◽  
Mohamed Kerada ◽  
Mehmet Acikgoz
Keyword(s):  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Fibonacci Numbers ◽  
Fibonacci Polynomials ◽  
Pell Numbers ◽  
New Class ◽  
Special Numbers And Polynomials ◽  
Symmetric Properties

In this paper, we derive some new symmetric properties of k-Fibonacci numbers by making use of symmetrizing operator. We also give some new generating functions for the products of some special numbers such as k-Fibonacci numbers, k-Pell numbers, Jacobsthal numbers, Fibonacci polynomials and Chebyshev polynomials.

scholarly journals 2-Fibonacci polynomials in the family of Fibonacci numbers

2018 ◽  
Vol 24 (3) ◽  
pp. 47-55 ◽  
Author(s):  
Engin Özkan ◽  
◽  
Merve Taştan ◽  
Ali Aydoğdu ◽  
◽  
...  
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Polynomials ◽  
The Family

scholarly journals (2, k)-Distance Fibonacci Polynomials

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 303
Author(s):  
Dorota Bród ◽  
Andrzej Włoch
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Polynomials ◽  
Pascal’S Triangle ◽  
Graph Interpretation ◽  
Natural Extensions ◽  
Pascal's Triangle

In this paper we introduce and study (2,k)-distance Fibonacci polynomials which are natural extensions of (2,k)-Fibonacci numbers. We give some properties of these polynomials—among others, a graph interpretation and matrix generators. Moreover, we present some connections of (2,k)-distance Fibonacci polynomials with Pascal’s triangle.

scholarly journals On Chebyshev Polynomials, Fibonacci Polynomials, and Their Derivatives

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Yang Li
Keyword(s):  
Chebyshev Polynomials ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Fibonacci Polynomials ◽  
Relationship Of ◽  
The Relationship ◽  
Derivatives Of

We study the relationship of the Chebyshev polynomials, Fibonacci polynomials, and theirrth derivatives. We get the formulas for therth derivatives of Chebyshev polynomials being represented by Chebyshev polynomials and Fibonacci polynomials. At last, we get several identities about the Fibonacci numbers and Lucas numbers.

scholarly journals Enumerative Combinatorics of Intervals in the Dyck Pattern Poset

Order ◽  
2021 ◽  
Author(s):  
Antonio Bernini ◽  
Matteo Cervetti ◽  
Luca Ferrari ◽  
Einar Steingrímsson
Keyword(s):  
Möbius Function ◽  
Dyck Path ◽  
Enumerative Combinatorics ◽  
Closed Expression ◽  
Mobius Function ◽  
First Results ◽  
Two Peaks

AbstractWe initiate the study of the enumerative combinatorics of the intervals in the Dyck pattern poset. More specifically, we find some closed formulas to express the size of some specific intervals, as well as the number of their covering relations. In most of the cases, we are also able to refine our formulas by rank. We also provide the first results on the Möbius function of the Dyck pattern poset, giving for instance a closed expression for the Möbius function of initial intervals whose maximum is a Dyck path having exactly two peaks.

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