On One-Parameter Generalization of Jacobsthal Numbers

In this paper, we discuss a one-parameter generalization of Jacobsthal numbers that preserves the recurrence relation with arbitrary initial conditions. We introduce generalized Jacobsthal-Lucas-like numbers, which are simple associations of generalized Jacobsthal numbers. Consequently, we give some new and well-known identities. Furthermore, we propose integral representations of these numbers associated with generalized Jacobsthal and Jacobsthal-Lucas-like numbers. Our results not only generalize the integral representations of the Jacobsthal and Jacobsthal-Lucas numbers but also apply to all one-parameter generalizations of Jacobsthal numbers.