Mersenne Lucas numbers and complete homogeneous symmetric functions

Abstract

In this paper, we first introduce new definition of Mersenne Lucas numbers sequence as, for n 2, m n = 3m n−1 − 2m n−2 with the initial conditions m 0 = 2 and m 1 = 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.

Ome new theorems on generating functions and their applications on odd and even certain numbers attached to p and q parameters

“…the famous examples of these sequences are (p, q)-Fibonacci and (p, q)-Lucas numbers, (p, q)-Jacobsthal and (p, q)-Jacobsthal Lucas numbers, (p, q)-Pell and (p, q)-Pell Lucas numbers (see [8,7,6,10,22,16,15]), because they are extensively used in various research areas. The authors in [14] defined the generalized (p, q)-Fibonacci sequence {f p,q,n (α, β, γ)} n≥0 , generalized (p, q)-Pell sequence {l p,q,n (α, β, γ)} n≥0 and generalized (p, q)-Jacobsthal sequence {C p,q,n (α, β, γ)} n≥0 as follows:…”

Some new properties of generalized ( p , q ) -numbers and generating functions for certain products

“…In modern science, there are a huge interest in (p, q)-numbers and thier properties in [8,10,14,18,19]. There are many generalizations of these numbers, the generalized (p, q)-Fibonacci numbers {f p,q,n (α, β, γ)} n∈N , generalized (p, q)-Pell numbers {l p,q,n (α, β, γ)} n∈N , and generalized (p, q)-Jacobsthal numbers {C p,q,n (α, β, γ)} n∈N [16] are one of them, f p,q,0 = α, f p,q,1 = β + γp, and f p,q,n = pf p,q,n−1 + qf p,q,n−2 , (…”

On The Hyperbolic k-Mersenne And k-Mersenne-Lucas Octonions

“…They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. Mersenne numbers have been studied in the literature and various generalizations such as Mersenne-Lucas, k-Mersenne, k-Mersenne-Lucas have been studied [1,4,6,7,17,22,[25][26][27].…”