The sequence of higher order Mersenne numbers and associated binomial transforms

TL;DR

This paper introduces the higher order Mersenne sequence, explores its algebraic properties, relations with other sequences, and examines its binomial transforms, matrix representations, and recurrence relations.

Contribution

It presents a new higher order Mersenne sequence, analyzes its properties, and studies its binomial transforms and matrix representations, extending the understanding of related integer sequences.

Findings

Derived Binet's formula for the sequence

Established recurrence relations and identities

Provided matrix and tridiagonal representations

Abstract

In this article, we introduce and study a new integer sequence referred to as the higher order Mersenne sequence. The proposed sequence is analogous to the higher order Fibonacci numbers and closely associated with the Mersenne numbers. Here, we discuss various algebraic properties such as Binet's formula, Catalan's identity, d'Ocagne's identity, generating functions, finite and binomial sums, etc. of this new sequence, and some inter-relations with Mersenne and Jacobsthal numbers. Moreover, we study the sequence generated from the binomial transforms of the higher order Mersenne numbers and present the recurrence relation and algebraic properties of them. Lastly, we give matrix generators and tridiagonal matrix representation for higher order Mersenne numbers.