scholarly journals On generalized Lucas polynomials and Euler numbers

2010 ◽  
Vol 11 (2) ◽  
pp. 163
Author(s):  
Ayşe Nalli ◽  
Tianping Zhang
Keyword(s):  
Euler Numbers ◽  
Lucas Polynomials ◽  
Generalized Lucas Polynomials

scholarly journals Lucas polynomials semi-analytic solution for fractional multi-term initial value problems

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Mahmoud M. Mokhtar ◽  
Amany S. Mohamed
Keyword(s):  
Initial Value Problems ◽  
Analytic Solution ◽  
Collocation Methods ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Fractional Differentiation ◽  
Initial Value ◽  
Lucas Polynomials ◽  
Error Analyses ◽  
Generalized Lucas Polynomials

AbstractHerein, we use the generalized Lucas polynomials to find an approximate numerical solution for fractional initial value problems (FIVPs). The method depends on the operational matrices for fractional differentiation and integration of generalized Lucas polynomials in the Caputo sense. We obtain these solutions using tau and collocation methods. We apply these methods by transforming the FIVP into systems of algebraic equations. The convergence and error analyses are discussed in detail. The applicability and efficiency of the method are tested and verified through numerical examples.

scholarly journals A Combinatorial Proof of a Result on Generalized Lucas Polynomials

2016 ◽  
Vol 49 (3) ◽  
Author(s):  
Alexandre Laugier ◽  
Manjil P. Saikia
Keyword(s):  
Recurrence Relation ◽  
Elementary Property ◽  
Combinatorial Proof ◽  
Lucas Polynomials ◽  
Initial Values ◽  
Generalized Lucas Polynomials

AbstractWe give a combinatorial proof of an elementary property of generalized Lucas polynomials, inspired by [1]. These polynomials in s and t are defined by the recurrence relation 〈n〉 = s〈n-1〉+t〈n-2〉 for n ≥ 2. The initial values are 〈0〉 = 2; 〈1〉= s, respectively.

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.

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