Integral representations of the Jacobsthal and Jacobsthal-Lucas numbers

Our study aims to obtain integral representations of Jacobsthal J_kn and Jacobsthal-Lucas L_kn, and then to use these integral representations to derive integral representations of Jacobsthal J_kn+r and Jacobsthal-Lucas L_kn+r, where n\in Z_0={0,1,2,..} is a non-negative integer, k \in Z_>0=1,2,3,... is an arbitrary but fixed positive integer, while r \in Z_0 is an arbitrary but fixed non-negative integer.