Generalized Jacobsthal-Narayana Numbers and Generalized Co-Jacobsthal-Narayana Numbers
Yuksel Soykan *
Department of Mathematics, Faculty of Science, Zonguldak B¨ ulent Ecevit University, 67100, Zonguldak, Turkey.
*Author to whom correspondence should be addressed.
Abstract
This paper introduces two innovative third-order recurrence sequences: the generalized Jacobsthal-Narayana sequence and the co-Jacobsthal-Narayana sequence. It examines their interrelated properties, including Binet’s formulas, generating functions, Simson’s formulas, and matrix representations, as well as their special subsequences. The study highlights unique relationships between recurrence equations and roots of characteristic equations, uncovering novel properties. These sequences hold promising potential for applications in various fields, such as number theory, combinatorics, cryptography, and modeling phenomena in physics, biology, and economics. For instance, the matrix representation of co-Jacobsthal-Narayana-Lucas numbers has implications for cryptographic systems.
Keywords: Jacobsthal-narayana numbers, jacobsthal-narayana-lucas numbers, co-jacobsthal-narayana numbers, co-jacobsthal-narayana-lucas numbers, third order recurrence relations, binet’s formula, generating functions