### On blocks of arithmetic progressions with equal products

Let $f\left(X\right)\in \mathbb{Q}\left[X\right]$ be a monic polynomial which is a power of a polynomial $g\left(X\right)\in \mathbb{Q}\left[X\right]$ of degree $\mu \ge 2$ and having simple real roots. For given positive integers ${d}_{1},{d}_{2},\ell ,m$ with $\ell \<m$ and gcd$(\ell ,m)=1$ with $\mu \le m+1$ whenever $m\<2$, we show that the equation $$f\left(x\right)f(x+{d}_{1})\cdots f(x+(\ell k-1){d}_{1})=f\left(y\right)f(y+{d}_{2})\cdots f(y+(mk-1){d}_{2})$$ with $f(x+j{d}_{1})\ne 0$ for $0\le j\<\ell k$ has only finitely many solutions in integers $x,y$ and $k\ge 1$ except in the case $$m=\mu =2,\ell =k={d}_{2}=1,f\left(X\right)=g\left(X\right),x=f\left(y\right)+y.$$