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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 givenintegers under natural conditions. We prove some effective results for specialvalues for $c,k$ and $\ell$ and obtain a general ineffective result based onBilu-Tichy method.

On equal values of products and power sums of consecutive elements in an arithmetic progression