  In this paper we study the Diophantine equation \begin{align*} b^k +
\left(a+b\right)^k + &\left(2a+b\right)^k + \ldots + \left(a\left(x-1\right) +
b\right)^k = \\ &y\left(y+c\right) \left(y+2c\right) \ldots \left(y+
\left(\ell-1\right)c\right), \end{align*} where $a,b,c,k,\ell$ are given
integers under natural conditions. We prove some effective results for special
values for $c,k$ and $\ell$ and obtain a general ineffective result based on
Bilu-Tichy method.
