Mersenne, jacobsthal, and jacobsthal-lucas numbers with negative subscripts

Abstract

In this paper, we extend the usual Mersenne, Jacobsthal, and Jacobsthal-Lucas numbers to their terms with negative subscripts. Many identities for new forms of these numbers, including Gelin-Cesáro identity, d’Ocagne’s identity, and some sum formulas are presented. Furthermore, we give certain generating matrices and show how the sums of the presented number sequences can be computed by employing matrix technique.