Higher-Order Mersenne Numbers: New Sequences, Algebraic Properties, and Binomial Transforms

Year 2025, Volume: 13 Issue: 4, 209 - 223, 15.12.2025

Kalika Prasad , Munesh Kumari , Rabiranjan Mohanta , Hrishikesh Mahato

https://doi.org/10.36753/mathenot.1797254

Abstract

This article introduces and investigates a new integer sequence, termed the higher-order Mersenne sequence, defined in analogy with higher-order Fibonacci numbers and closely related to classical Mersenne numbers. We establish a range of fundamental algebraic properties of this sequence, including its Binet-type formula, Catalan’s identity, d’Ocagne’s identity, generating function, and several finite and binomial summation identities. Further, we explore its connections with both Mersenne and Jacobsthal numbers. The study also examines the sequence obtained via the binomial transform of higher-order Mersenne numbers, deriving its recurrence relation and algebraic characteristics. In addition, matrix generators and a tridiagonal matrix representation are developed to enrich the structural understanding of these numbers.

Keywords

Binomial sums , Binomial transform , Generating functions , Matrix generators , Mersenne numbers , Partial sums , Recurrence relations

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