Synthesizing Sums

Polynomial functions $f : \mathbb{N}_+ \longrightarrow \mathbb{N}_+$ are studied for which sums of arbitrary length $f (1) + f (2) + f (3) + >... + f (n)$, with $n \in \mathbb{N}_+$, can be expressed by polynomial functions $g : \mathbb{N}_+ \longrightarrow \mathbb{N}_+$ which involve a bounded number of operations, thus not depending on $n$. Open problems and extensions are presented.