Some new properties of modified Jacobsthal and Jacobsthal-Lucas numbers

Abstract

A certain generalization of Jacobsthal numbers was proposed in the form Jns,t = sn-(t)ns+t, where n ≥ 0 is a natural number and s ≠ -t are arbitrary real numbers (Atanassov 2011). As an analogue, a modification of Jacobsthal-Lucas numbers was formulated in the form jns,t = sn+(-t)n, where n is a natural number and s and t are arbitrary real numbers (Shang 2012). In fact, these modifications can be considered as certain generalizations of Fibonacci and Lucas numbers. Now, it appears that only few have studied these modifications (e.g. Rabago, 2013), at least we have not seen related papers before. Hence, we investigate some of their properties and obtain several identities using matrices. We also prove a general d'Ocagne's identity using a new approach.