Distributions and Fibonacci Polynomials of Order k, Longest Runs, and Reliability of Consecutive-k-out-of-n : F Systems

1986 ◽  
pp. 203-227 ◽  
Author(s):  
Andreas N. Philippou
Keyword(s):  
Fibonacci Polynomials

scholarly journals Elliptic Solutions of Dynamical Lucas Sequences

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 183
Author(s):  
Michael J. Schlosser ◽  
Meesue Yoo
Keyword(s):  
Discrete Time ◽  
Time Dependent ◽  
Fibonacci Polynomials ◽  
Lucas Sequences ◽  
Elliptic Solutions

We study two types of dynamical extensions of Lucas sequences and give elliptic solutions for them. The first type concerns a level-dependent (or discrete time-dependent) version involving commuting variables. We show that a nice solution for this system is given by elliptic numbers. The second type involves a non-commutative version of Lucas sequences which defines the non-commutative (or abstract) Fibonacci polynomials introduced by Johann Cigler. If the non-commuting variables are specialized to be elliptic-commuting variables the abstract Fibonacci polynomials become non-commutative elliptic Fibonacci polynomials. Some properties we derive for these include their explicit expansion in terms of normalized monomials and a non-commutative elliptic Euler–Cassini identity.

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 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 Alternative proofs of some formulas for two tridiagonal determinants

2018 ◽  
Vol 10 (2) ◽  
pp. 287-297 ◽  
Author(s):  
Feng Qi ◽  
Ai-Qi Liu
Keyword(s):  
Chebyshev Polynomials ◽  
Tridiagonal Matrix ◽  
Fibonacci Polynomials ◽  
Detailed Proof ◽  
Delannoy Numbers ◽  
Appl Math ◽  
Math Phys

Abstract In the paper, the authors provide five alternative proofs of two formulas for a tridiagonal determinant, supply a detailed proof of the inverse of the corresponding tridiagonal matrix, and provide a proof for a formula of another tridiagonal determinant. This is a companion of the paper [F. Qi, V. Čerňanová,and Y. S. Semenov, Some tridiagonal determinants related to central Delannoy numbers, the Chebyshev polynomials, and the Fibonacci polynomials, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 81 (2019), in press.

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