Extension of Erdos-Selfridge Theorem

Question

Erdos and Selfridge open their paper "The Product of Consecutive Integers is Never a Power" (1974) with the theorem

$\text{Theorem 1:}$ The product of two or more consecutive positive integers is never a power.

Question: While I am working through the paper to understand their argument, I wonder: has there been any work extending these results to other rings of integers? i.e. falsifying the equation in general $$ (a+1)\cdots(a+k)=b^l $$ for (what I suspect) some number field $F$, positive $a,b\in\mathcal{O}_F$, and $k,l\geq 2$.

Note: I know that for $F=\mathbb{Q}[\sqrt{5}],\>$ $a=b=\frac{1}{2}(\sqrt{5}-1),\>k=2,\>l=3$, the equation is true. Perhaps it would invite a reason to restrict the question to fields $F=\mathbb{Q}[\sqrt{m}]$ where $m\not\equiv 1\mod 4.$

Answer 1

It's not hard to find number fields where there are examples. E.g., the equation $(y-3)(y-2)(y-1)=y^2$, equivalently $y^3-7y^2+11y-6=0$, has a root $\alpha$, an algebraic integer, with $5<\alpha<6$, so if we let $x=\alpha-4$, then $x$ is a positive algebraic integer with $(x+1)(x+2)(x+3)=(x+4)^2$.