On Some Identities and Generating Functions for Mersenne Numbers and Polynomials

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.


Introduction
In this paper we consider one of the sequences of positive integers satisfying a recurrence relation and we give some well-known identities for this type of sequences [9]. One of the sequences of positive integers (also defined recursively) that have been studied over several years is the well-known Fibonacci (and Lucas) sequence. Many papers are dedicated to Fibonacci sequence, such as the works of Hoggatt in [10], and Koshy in [11], among others. Others sequences satisfying a second-order recurrence relations are the main topic of the research for several authors, such as the studies of the sequences N { } n n J ∈ and N { } n n j ∈ of Jacobsthal and Jacobsthal-Lucas numbers, respectively.
In this paper we do not have such kind of generalization, but we will follow closely some of these studies. About the Mersenne sequence, also some studies about this sequence have been published, such as the work of Koshy [12], where the authors investigate some divisibility properties of Catalan numbers with Mersenne numbers Mk as their subscripts, developing their work in [12]. In number theory, recall that a Mersenne number of order n, denoted by Mn, is a number of the form 2 1 n − , where n is a nonnegative number. This identity is called as the Binet formula for Mersenne sequence and it comes from the fact that the Mersenne numbers can also be defined recursively by 1    The main purpose of this paper is to present some results involving the Mersenne numbers using define a new useful operator denoted by 1 for which we can formulate, extend and prove new results based on our previous ones [1,3,4]. In order to determine generating functions of the product of Mersenne numbers and Chebychev polynomials of first and second kind, we combine between our indicated past techniques and these presented polishing approaches.

Definitions and Some Properties
In order to render the work self-contained we give the necessary preliminaries tools; we recall some definitions and results. Definition1. [6] Let B and P be any two alphabets. We define ( ) n S B P − by the following form We know that the polynomial whose roots are P is written as On the other hand, if B has cardinality equal to 1, i.e.,

{ },
can be rewritten as follows [6]: The summation is actually limited to a finite number of terms since Given a function f on R n , the divided difference operator is defined as follows 1 1

On the Generating Functions
In our main result, we will combine all these results in a unified way such that they can be considered as a special case of the following Theorem.
for all . k ∈ Proof By applying the operator Thus, this completes the proof. We now derive new generating functions of the products of some well-known polynomials. Indeed, we consider Theorem 1 in order to derive Mersenne numbers and Tchebychev polynomials of second kind and the symmetric functions • If k=0 and A={1,0} we deduce the following lemma Lemma [1] Given an alphabet { }