Putnam 2019 B5

Let $F_m$ be the $m$th Fibonacci number, defined by $F_1 = F_2 = 1$ and $F_m = F_{m-1} + F_{m-2}$ for all $m \geq 3$. Let $p(x)$ be the polynomial of degree $1008$ such that $p(2n+1) = F_{2n+1}$ for $n=0,1,2,\dots,1008$. Find integers $j$ and $k$ such that $p(2019) = F_j - F_k$.