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

Kalika Prasad; Munesh Kumari; Rabiranjan Mohanta; Hrishikesh Mahato

doi:10.36753/mathenot.1797254

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

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

References

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Details

Primary Language

English

Subjects

Mathematical Methods and Special Functions, Complex Systems in Mathematics, Approximation Theory and Asymptotic Methods, Applied Mathematics (Other)

Journal Section

Research Article

Authors

Kalika Prasad
0000-0002-3653-5854
India

Munesh Kumari ^*
0000-0002-6541-0284
India

Rabiranjan Mohanta
0009-0002-9197-3539
India

Hrishikesh Mahato
0000-0002-3769-0653
India

Early Pub Date

December 9, 2025

Publication Date

December 15, 2025

Submission Date

October 5, 2025

Acceptance Date

November 30, 2025

Published in Issue

Year 2025 Volume: 13 Number: 4

DOI

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

IZ

https://izlik.org/JA76TB38EK

APA

Prasad, K., Kumari, M., Mohanta, R., & Mahato, H. (2025). Higher-Order Mersenne Numbers: New Sequences, Algebraic Properties, and Binomial Transforms. Mathematical Sciences and Applications E-Notes, 13(4), 209-223. https://doi.org/10.36753/mathenot.1797254

AMA

1.Prasad K, Kumari M, Mohanta R, Mahato H. Higher-Order Mersenne Numbers: New Sequences, Algebraic Properties, and Binomial Transforms. Math. Sci. Appl. E-Notes. 2025;13(4):209-223. doi:10.36753/mathenot.1797254

Chicago

Prasad, Kalika, Munesh Kumari, Rabiranjan Mohanta, and Hrishikesh Mahato. 2025. “Higher-Order Mersenne Numbers: New Sequences, Algebraic Properties, and Binomial Transforms”. Mathematical Sciences and Applications E-Notes 13 (4): 209-23. https://doi.org/10.36753/mathenot.1797254.

EndNote

Prasad K, Kumari M, Mohanta R, Mahato H (December 1, 2025) Higher-Order Mersenne Numbers: New Sequences, Algebraic Properties, and Binomial Transforms. Mathematical Sciences and Applications E-Notes 13 4 209–223.

IEEE

[1]K. Prasad, M. Kumari, R. Mohanta, and H. Mahato, “Higher-Order Mersenne Numbers: New Sequences, Algebraic Properties, and Binomial Transforms”, Math. Sci. Appl. E-Notes, vol. 13, no. 4, pp. 209–223, Dec. 2025, doi: 10.36753/mathenot.1797254.

ISNAD

Prasad, Kalika - Kumari, Munesh - Mohanta, Rabiranjan - Mahato, Hrishikesh. “Higher-Order Mersenne Numbers: New Sequences, Algebraic Properties, and Binomial Transforms”. Mathematical Sciences and Applications E-Notes 13/4 (December 1, 2025): 209-223. https://doi.org/10.36753/mathenot.1797254.

JAMA

1.Prasad K, Kumari M, Mohanta R, Mahato H. Higher-Order Mersenne Numbers: New Sequences, Algebraic Properties, and Binomial Transforms. Math. Sci. Appl. E-Notes. 2025;13:209–223.

MLA

Prasad, Kalika, et al. “Higher-Order Mersenne Numbers: New Sequences, Algebraic Properties, and Binomial Transforms”. Mathematical Sciences and Applications E-Notes, vol. 13, no. 4, Dec. 2025, pp. 209-23, doi:10.36753/mathenot.1797254.

Vancouver

1.Kalika Prasad, Munesh Kumari, Rabiranjan Mohanta, Hrishikesh Mahato. Higher-Order Mersenne Numbers: New Sequences, Algebraic Properties, and Binomial Transforms. Math. Sci. Appl. E-Notes. 2025 Dec. 1;13(4):209-23. doi:10.36753/mathenot.1797254