Effective results for polynomial values of (alternating) power sums of arithmetic progressions

We prove effective finiteness results concerning polynomial values of the sums $$b^k+{(a+b)}^k+\cdots +{(a(x-1)+b)}^k$$ and $$b^k-{(a+b)}^k+{(2a+b)}^k-\dots +(-1{)}^{x-1}{(a(x-1)+b)}^k,$$ where $a\mathrm{\ne }0,b,k$ are given integers with $\gcd (a,b)=1$ and $k\ge 2$.