This is referring to Stanley's Enumerative Combinatorics book, Chapter 1 Ex6.
For $n \in \mathbb{Z}$ define $J_{n}(2x)=\sum_{k \in \mathbb{Z}} \frac{(-1)^kx^{n+2k}}{k!(n+k)!}$ Let $\frac{1}{j!}=0$ if $j<0$. Show that $e^x = \sum_{n≥0} L_nJ_n(2x)$ where $L_n$ is the Lucas number.
Edit: I've tried to show that RHS satisfies the property that $\frac{d}{dx}\sum_{n≥0} L_nJ_n(2x) = \sum_{n≥0} L_nJ_n(2x)$ but it gets messy. I've also tried to use $f(a+b) = f(a)f(b)$ on RHS but that doesn't seem to work either.