On some properties of Jacobsthal and Jacobsthal-Lucas numbers: Some properties of Jacobsthal and Jacobsthal-Lucas numbers

Abstract

In this study, we first develop some helpful generating functions for the Jacobsthal and Jacobsthal-Lucas numbers. We have discussed generating functions for $\mathbf{J}_{2n}$ and $\mathbf{j}_{2n}$ with just even and odd terms and generating functions for $\mathbf{J}_{2n+1}$, $\mathbf{j}_{2n+1}$, $\mathbf{J}_n\mathbf{J}_{n+1}$, $\mathbf{j}_n\mathbf{j}_{n+1}$, $\mathbf{J}_n^2$ and $\mathbf{j}_n^2$, where $\mathbf{J}_n$ and $\mathbf{j}_n$ are $n^{th}$ Jacobsthal and Jacobsthal-Lucas numbers, respectively. Using these generating functions, we have found sum formulas for square Jacobsthal and Jacobsthal-Lucas numbers. Finally, we derive numerous useful convolution formulas including Jacobsthal numbers and similar expressions for Jacobsthal-Lucas numbers.