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

Abstract

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