The sequence of higher order Mersenne numbers and associated binomial transforms

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.