The sequence of higher order Mersenne numbers and associated binomial transforms
Kalika Prasad, Munesh Kumari, Rabiranjan Mohanta, Hrishikesh Mahato

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.
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Taxonomy
TopicsAdvanced Mathematical Theories and Applications · Advanced Mathematical Theories · Fractal and DNA sequence analysis
