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

In this paper, we study the Diophantine equation $$ b^k + \left(a+b\right)^k + \left(2a+b\right)^k + \cdots + \left(a\left(x-1\right) + b\right)^k = y\left(y+c\right) \left(y+2c\right) \cdots \left(y+ \left(\ell-1\right)c\right), $$ 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 the Bilu—Tichy method.