Matrix Representation of Bi-Periodic Jacobsthal Sequence

TL;DR

This paper introduces a matrix representation for the bi-periodic Jacobsthal sequence, deriving its general term, properties, generating function, Binet formula, and new summation formulas, expanding understanding of this sequence.

Contribution

The paper presents the first matrix representation of the bi-periodic Jacobsthal sequence and derives its key properties and formulas, including the Binet and generating functions.

Findings

Derived the nth general term of the matrix sequence

Obtained the Cassini or Simpson's formula for the sequence

Presented new summation formulas and properties

Abstract

In this paper, we bring into light the matrix representation of bi-periodic Jacobsthal sequence, which we shall call the bi-periodic Jacobsthal Matrix sequence.. We obtained the nth general term of this new matrix sequence. By studying the properties of this new matrix sequence, the well-known Cassini or Simpson's formula was obtained. We then proceeded to find its generating function as well as the Binet formula. Some new properties and two summation formulas for this new generalized matrix sequence are also given. Keywords: Bi-periodic Jacobsthal sequence; Generating function; Binet formula.