Mersenne Lucas numbers and complete homogeneous symmetric functions

Abstract

In this paper, we first introduce new definition of Mersenne Lucas numbers sequence as, for \(n\geq 2,\) \(m_{n}=3m_{n-1}-2m_{n-2}\) 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 \(% \left( p,q\right) \)-numbers with Mersenne Lucas numbers at positive and negative indice.