Weighted sums of consecutive values of a polynomial

In this paper, we determine all sets A of integers such that, for any integral-valued polynomial f(x) which has no fixed divisor, for all integers l⩾1 and n, there are infinitely many integers m>l and a choice of εi∈A such that n=εlf(l)+εl+1f(l+1)+⋯+εmf(m). The earlier result shows that A={−1,1} is such a set.