On the Generalized Order-k Jacobsthal and Jacobtshal-Lucas Numbers - Daşdemir, Ahmet; Bilgici, Göksal; Ahmed, Hossen (Fundamentals of Contemporary Mathematical Sciences, 2025)
The classic Jacobsthal numbers were generalized to k sequences of the generalized order-k Jacobsthal numbers and then have been studied by several authors. In this paper, we explain that all of these studies used an incorrect version of order-k Jacobsthal numbers for reasons and give the correct definition of order-k Jacobsthal numbers. Further, we introduce the compatible generalized order-k Jacobsthal-Lucas numbers with the generalized order-k Jacobsthal numbers. Next, we give some properties of order-k Jacobsthal numbers and order-k Jacobsthal-Lucas numbers, including generating matrix, generalized Binet’s formula, and elementary matrix identities. Further, we investigate specific examples for our results and give special identities, i.e., sum formula and interrelationships between these sequences.
Keywords: Algebra and Number Theory; Cebir ve Sayı Teorisi; Generalized order-k sequence; Jacobsthal sequence; trace of matrix; Binet; Jacobsthal-Lucas sequence.