Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a general method to construct identities of number or po...

The focus of this paper is to study the HOMFLY polynomial of (2, n)-torus link as a generalized Fibonacci polynomial. For this purpose, we first introduce a form of generalized Fibonacci and Lucas polynomials and provide their some fundamental properties. We define the HOMFLY polynomial of (2, n)-torus link with a way similar to our generalized Fibonacci polynomials and provide its fundamental ...

We deduce exact formulas for polynomials representing the Lucas logarithm and prove lower bounds on the degree of interpolation polynomials of the Lucas logarithm for subsets of given data. © 2005 Elsevier Inc. All rights reserved.

In this article, we introduce a family of weighted lattice paths, whose step set is {H = (1, 0), V = (0, 1), D1 = (1, 1), . . . , Dm−1 = (1,m − 1)}. Using these lattice paths, we define a family of Riordan arrays whose sum on the rising diagonal is the k-bonacci sequence. This construction generalizes the Pascal and Delannoy Riordan arrays, whose sum on the rising diagonal is the Fibonacci and ...

Abstract The major goal of this research is to develop and test a numerical technique for solving linear one-dimensional telegraph problem. generalized polynomials, namely, the Lucas polynomials are selected as basis functions. To solve type equation, we instead its corresponding integral equation via application spectral Galerkin method that serves convert with underlying conditions into syste...

Recently, Belbachir and Bencherif have expanded Fibonacci and Lucas polynomials using bases of Fibonacci-and Lucas-like polynomials. Here, we provide simplified proofs for the expansion formulaethat in essence a computer can do. Furthermore, for 2 of the 5 instances, we find q-analogues.

This paper is concerned with the generalized Euler polynomial matrix $\E^{(\alpha)}(x)$ and $\E$. Taking into account some properties of polynomials numbers, we deduce product formulae for determine inverse We establish explicit expressions $\E(x)$, which involving Pascal, Fibonacci Lucas matrices, respectively. From these get new interesting identities numbers. Also, provide factorizations in ...

For a nonnegative integer p, we give explicit formulas for the p-Frobenius number and p-genus of generalized Fibonacci numerical semigroups. Here, p-numerical semigroup Sp is defined as set integers whose integral linear combinations given positive a1,a2,…,ak are expressed in more than p ways. When p=0, S0 with 0-Frobenius 0-genus original Frobenius genus. In this paper, consider involving Jaco...

In this paper, we consider fields determined by the /1 roots of the zeros a and fi of the polynomial x x 1 ; a is the positive zero. The tools for studying these fields will include the Fibonacci and Lucas polynomials. Generalized versions of Fibonacci and Lucas polynomials have been studied in [1], [2], [3], [4], [5], [6], [7], and [12], among others. For the most part, these generalizations c...