In this paper, we will introduce a new definition of k-Mersenne–Lucas numbers and investigate some properties. Then, we obtain some identities and established connection formulas between k-Mersenne–Lucas numbers and k-Mersenne numbers through the use of Binet’s formula.

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On the k-Mersenne–Lucas numbers

The main aim of this paper is to introduce new concept of bivariate Mersenne Lucas polynomials { m n ( x , y ) } n = 0 ∞ , we first give the recurrence relation of them. We then obtain Binet’s formula, generating function, Catalan’s identity and Cassini’s identity for this type of polynomials. After that, we give the symmetric function, explicit formula and d’Ocagne’s identity of bivariate Mersenne and bivariate Mersenne Lucas polynomials. By using the Binet’s formula we obtain some well-known identities of these bivariate polynomials. Also, some summation formulas of bivariate Mersenne and bivariate Mersenne Lucas polynomials are investigated.

On the bivariate Mersenne Lucas polynomials and their properties

This paper aims to introduce the k -Mersenne and k -Mersenne-Lucas octonions. We give the closed form formulae for these octonions and obtain some well-known identities like Cassini’s identity, d’Ocagne’s identity, Catalan identity, Vajda’s identity and generating functions of them. As a consequence k = 1 yields all the above properties for Mersenne and Mersenne-Lucas octonions.

On $k$-Mersenne and k-Mersenne-Lucas Octonions

In this paper, we study the symmetric and the generating functions for odd
 and even terms of the second-order linear recurrence sequences. we introduce
 a operator in order to derive a new family of generating functions of odd
 and even terms of Mersenne numbers, Mersenne Lucas numbers, (p,q)-
 Fibonacci-like numbers, k-Pell polynomials and k-Pell Lucas polynomials. By
 making use of the operator defined in this paper, we give some new
 generating functions of the products of (p,q)-Fibonacci-like numbers
 with odd and even terms of certain numbers and polynomials.

/pdf/generating-functions-of-binary-products-of-p-q-fibonacci-1bfculrt.pdf

Generating functions of binary products of (p,q)-Fibonacci-like numbers with odd and even certain numbers and polynomials

A bstract . In this paper, we introduce a generalized family of numbers and polynomials of one or more variables attached to the formal composition f . ( g ◦ h ) of generating functions f , g and h . We give explicit formula and apply the obtained result to two special families of polynomials; the ﬁrst concerns generalization of some polynomials applied to the theory of hyperbolic differential equations recently introduced and studied by M. Mihoubi and M. Sahari. The second concerns two variables Laguerre-based generalized Hermite-Euler polynomials introduced and should be updated to studied recently by N. U. Khan et al..

On polynomials associated to product and composition of generating functions

In this paper, we first introduce new definition of Mersenne Lucas numbers sequence as, for n > 2, mn = 3mn−1 − 2mn−2 with the initial conditions m0 = 2 and m1 = 3. Considering this sequence, we give Binet’s formula, generating function and symmetric function of Mersenne Lucas numbers. By using the Binet’s formula we obtain some well-known identities such as Catalan’s identity, Cassini’s identity and d’Ocagne’s identity. After that, we give some new generating functions for products of (p,q)-numbers with Mersenne Lucas numbers at positive and negative indice.

/pdf/mersenne-lucas-numbers-and-complete-homogeneous-symmetric-4w6i95dmct.pdf

Kasi Viswanadh V. Kanuri

Papers

Mersenne Lucas numbers and complete homogeneous symmetric functions