Generalized Fibonacci-Lucas Polynomials (original) (raw)
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On generalized Fibonacci and Lucas polynomials
Chaos, Solitons & Fractals, 2009
Let hðxÞ be a polynomial with real coefficients. We introduce hðxÞ-Fibonacci polynomials that generalize both Catalan's Fibonacci polynomials and Byrd's Fibonacci polynomials and also the k-Fibonacci numbers, and we provide properties for these hðxÞ-Fibonacci polynomials. We also introduce hðxÞ-Lucas polynomials that generalize the Lucas polynomials and present properties of these polynomials. In the last section we introduce the matrix Q h ðxÞ that generalizes the Q-matrix 1 1 1 0 whose powers generate the Fibonacci numbers.
The generalized k-Fibonacci polynomials and generalized k-Lucas polynomials
Notes on Number Theory and Discrete Mathematics, 2021
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.
Generalized Fibonacci-Like Polynomials and Some Identities
Global Journal of Mathematical Analysis, 2014
The Fibonacci polynomials and Lucas polynomials are famous for possessing wonderful and amazing properties and identities. In this paper, Generalized Fibonacci-Like Polynomials are introduced and defined by 1 2 () () (), n 2. n n n m x xm x m x with 0 () 2 m x s and 1 () 1 m x s , where s is integer. Further, some basic identities are generated and derived by standard methods.
Identities for the Generalized Fibonacci Polynomial
Integers, 2018
A second order polynomial sequence is of Fibonacci type (Lucas type) if its Binet formula is similar in structure to the Binet formula for the Fibonacci (Lucas) numbers. In this paper we generalize identities from Fibonacci numbers and Lucas numbers to Fibonacci type and Lucas type polynomials. A Fibonacci type polynomial is equivalent to a Lucas type polynomial if they both satisfy the same recurrence relations. Most of the identities provide relationships between two equivalent polynomials. In particular, each type of identities in this paper relate the following polynomial sequences: Fibonacci with Lucas, Pell with Pell-Lucas, Fermat with Fermat-Lucas, both types of Chebyshev polynomials, Jacobsthal with Jacobsthal-Lucas and both types of Morgan-Voyce.
Generalized Fibonacci – Lucas sequence its Properties
Global Journal of Mathematical Analysis, 2014
Sequences have been fascinating topic for mathematicians for centuries. The Fibonacci sequences are a source of many nice and interesting identities. A similar interpretation exists for Lucas sequence. The Fibonacci number, Lucas numbers and their generalization have many interesting properties and applications to almost every field. Fibonacci sequence is defined by the recurrence formula F F F n 2 n n-1 n-2 , and F 0,F 1 01 , where F n are an n th number of sequences. The Lucas Sequence is defined by the recurrence formula L L L n 2 n n-1 n-2 , and L =2, L =1 01 , where L n an nth number of sequences are. In this paper, we present generalized Fibonacci-Lucas sequence that is defined by the recurrence relation 12 B B B n n n , 2 n with B 0 = 2s, B 1 = s. We present some standard identities and determinant identities of generalized Fibonacci-Lucas sequences by Binet's formula and other simple methods.
A new approach to generalized Fibonacci and Lucas numbers with binomial coefficients
Applied Mathematics and Computation, 2013
In this study, Fibonacci and Lucas numbers have been obtained by using generalized Fibonacci numbers. In addition, some new properties of generalized Fibonacci numbers with binomial coefficients have been investigated to write generalized Fibonacci sequences in a new direct way. Furthermore, it has been given a new formula for some Lucas numbers.
Generalized Fibonacci-Lucas Sequence
Turkish Journal of Analysis and Number Theory, 2014
The Fibonacci sequence is a source of many nice and interesting identities. A similar interpretation exists for Lucas sequence. The Fibonacci sequence, Lucas numbers and their generalization have many interesting properties and applications to almost every field. Fibonacci sequence is defined by the recurrence formula − − = + , 2 n ≥ with B 0 = 2b, B 1 = s, where b and s are integers. We present some standard identities and determinant identities of generalized Fibonacci-Lucas sequences by Binet's formula and other simple methods.
Mathematics
The goal of this study is to develop some new connection formulae between two generalized classes of Fibonacci and Lucas polynomials. Hypergeometric functions of the kind 2F1(z) are included in all connection coefficients for a specific z. Several new connection formulae between some famous polynomials, such as Fibonacci, Lucas, Pell, Fermat, Pell–Lucas, and Fermat–Lucas polynomials, are deduced as special cases of the derived connection formulae. Some of the introduced formulae generalize some of those existing in the literature. As two applications of the derived connection formulae, some new formulae linking some celebrated numbers are given and also some newly closed formulae of certain definite weighted integrals are deduced. Based on using the two generalized classes of Fibonacci and Lucas polynomials, some new reduction formulae of certain odd and even radicals are developed.
arxiv.org, 2020
This paper is concerned with developing some new connection formulae between two generalized classes of Fibonacci and Lucas polynomials. All the connection coefficients involve hypergeometric functions of the type 2 F 1 (z), for certain z. Several new connection formulae between some famous polynomials such as Fibonacci, Lucas, Pell, Fermat, Pell-Lucas, and Fermat-Lucas polynomials are deduced as special cases of the derived connection formulae. Some of the introduced formulae generalize some of those existing in the literature. As two applications of the derived connection formulae, some new formulae linking some celebrated numbers are given and also some newly closed formulae of certain definite weighted integrals are deduced. Based on using the two generalized classes of Fibonacci and Lucas polynomials, some new reduction formulae of certain odd and even radicals are developed.
Fibonacci-Like Polynomials and Some Properties
International Journal of Advanced Mathematical Sciences, 2013
The Fibonacci sequence and Fibonacci polynomials are famous for possessing wonderful and amazing properties and identities. In this paper, Fibonacci-Like polynomials (FLP) is introduce and define by 12 2 n n n s (x) xs (x) s (x), n with 0 s (x)=2 and 1 s (x)=2x. Also some basic identities are presented and derived by standard method.