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.

1. 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 and 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 , 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

(1.1)

with initial conditions Since this recurrence is inhomogeneous, substituting n by n+1, we obtain the new form

(1.2)

Subtracting (1.1) to (1.2), we have that

and then

other form for the recurrence relation of Mersenne sequence, with initial conditions . The roots of the respective characteristic equation

are

and we easily get the Binet formula

The main purpose of this paper is to present some results involving the Mersenne numbers using define a new useful operator denoted by 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.

2. 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 by the following form

(2.1)

with the condition for

Equation (2.1) can be rewritten in the following form

where

(2.2)

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., then (2.1) can be rewritten as follows ⁶:

where for all

The summation is actually limited to a finite number of terms since for all In particular, we have

where are the coefficients of the polynomials for These coefficients are zero for

For example, if all are equal, i.e., then we have

By choosing P=1 i.e., we obtain

(2.3)

By combining (2.2) and (2.3), we obtain the following expression

Definition 2. ⁴ Given a function on Rⁿ, the divided difference operator is defined as follows

Definition 3. The symmetrizing operator is defined by

(2.4)

Proposition 1. ⁵ Let an alphabet, we define the operator as follows

3. 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.

Theorem 1. Let A and P be two alphabets, respectively, and then we have

(3.1)

for all

Proof By applying the operator to the series we have

On the other hand,

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 we have

(3.2)

Replacing by in (3.2), we obtain

(3.3)

Choosing and such that

and substituting in (3.3) we end up with

which represents a generating function for Mersenne numbers, such that

If and we deduce the following theorems

Theorem 2. ⁷ Given two alphabets and we have

(3.4)

Theorem 3. ⁸ Given two alphabets and we have

(3.5)

From (3.5) we can deduce

(3.6)

Case 1: Replacing by and by in (3.4) and (3.6) yields

(3.7)

(3.8)

This case consists of four related parts.

Firstly, the substitutions of

in (3.7) give

which represents a new generating function for product of Fibonacci numbers with Mersenne numbers, such that

Secondly, the substitution of

in (3.8) give

which represents a new generating function for Mersenne numbers of second order, such that

Thirdly, the substitution of

in (3.8) give

which represents a new generating function for product of Jacobsthal numbers with Mersenne numbers, such that

Finally, the substitution of

in (3.8) give

which represents a new generating function for product of Pell numbers with Mersenne numbers, such that

Case 2: Replacing by and by and by in (3.4) yields

(3.9)

The substitution of

in (3.9) and set for ease on notations we reach

which corresponds to a new generating function for the combined Mersenne numbers and Tchebychev polynomials of the second kind.

Theorem 4. For the new generating function of the product of Mersenne numbers and Tchebychev polynomials of first kind is given by

Proof We see that

On the other hand, we know that

from which it follows

Therefore

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions.

References