Polynomial values of sums of products of consecutive integers (Q1673734)

Following \textit{L. Hajdu} et al. [Acta Arith. 172, No. 4, 333--349 (2016; Zbl 1400.11087)], for \(k=0,1,2,\ldots \) let \[f_k(x)=\sum_{i=0}^k\prod_{j=0}^i (x+j).\] Let \(g(x)\) be an arbitrary polynomial with rational coefficients. Consider the equation \[f_k(x)=g(y)\;\; \text{in} \;\;\ x,y\in\mathbb Z.\] In case the degree of \(g\) is 0 or 2, the authors show that an effectively computable upper bound can be derived for the absolute values of the solutions. If the degree of \(g\) is \(\geq 3\), then it is shown, that the equation has only finitely many solutions, unless \(g(x)=f_k(h(x))\) with a polynomial \(h\) with rational coefficients and positive degree. In the proof of the second statement an ineffective theorem of \textit{Y. F. Bilu} and \textit{R. F. Tichy} [Acta Arith. 95, No. 3, 261--288 (2000; Zbl 0958.11049)] is applied.