On Some Identities and Generating Functions for Mersenne Numbers and Polynomials

Abstract

In this paper, we introduce a new operator in order to derive some properties of homogeneous symmetric functions. By making use of the proposed operator, we give some new generating functions for Mersenne numbers, Mersenne numbers and product of sequences and Chebychev polynomials of second kind.

Symmetric Functions for <i>k</i>-Fibonacci Numbers and Orthogonal Polynomials

“…The main purpose of this paper is to present some results involving the k-Fibonacci and k-Jacobsthal numbers using define a new useful operator denoted by δ By making use of this operator, we can derive new results based on our previous ones [8,9,10,11,12]. In order to determine generating functions of the product of k-Fibonacci and k-Jacobsthal numbers and Chebychev polynomials of second kind, we combine between our indicated past techniques and these presented polishing approaches.…”

Generating Functions of Binary Products of Tribonacci and Tribonacci Lucas Polynomials and Special Numbers

“…In this section, we introduce a new symmetric function and give some properties of this symmetric function [1,2,4,5,6]. We also give some more useful definitions from the literature which are used in the subsequent sections [8,9,10,16].…”

“…In this contribution, we shall define a new useful operator denoted by δ k e 1 e 2 for which we can formulate, extend and prove new results based on our previous ones, see [16], [3], [18]. In order to determine generating functions of product of k-Fibonacci, k-Lucas, k-Pell, k-Jacobsthal and Mersenne numbers with Tribonacci and Tribonacci Lucas polynomials, we combine between our indicated past techniques and these presented polishing approaches.…”

Construction of Symmetric Functions of Generalized Fibonacci Numbers

“…The following lemmas allows us to obtain many generating functions of generalized Fibonacci numbers and some well-known numbers cited above, using a technique symmetric functions. we refer the reader to see the references [4][5][6][7][8][9][10][11][12] .…”