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

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