Assume that $\alpha , n,\lambda \in \mathbb{N}$,and $f$,$g$ two real valued functions defined on $\mathbb{N}$. Function $W$ is given by the following formula.
\begin{equation} W(n) = \max_{1 \leq \lambda \leq n} \bigg\{ \sum^{\min\{\lambda + \alpha , n \}}_{k=\lambda} f(k) + \sum^{n}_{k=\min\{\lambda + \alpha , n \}+1}g(k) \bigg\} \end{equation}
I am trying to figure out whether it is possible to write this expression into a more compact recursive form, like writing $W(n)$ as a function of $W(n-1)$ and maybe some other terms. Is it possible? I can't figure out how to make $n-1$ appear and get a nice recursion, since $n$ is inside the minimum. Any help is greatly appreciated. Thanks!