On the equation \(x(x+d_ 1)\dots (x+(k-1)d_ 1)= y(y+d_ 2)\dots (y+(mk-1)d_ 2)\) (Q1322357)

For positive integers \(m \geq 2\), \(d_ 1\) and \(d_ 2\), the title equation is considered for unknown positive integers \(x,y\) and \(k \geq 2\). When \(d_ 1=d_ 2\) this equation reduces to the one investigated by the same authors in [Indag. Math. 3, 79-90 (1992; Zbl 0757.11011); ibid. 3, 237-242 (1992; Zbl 0763.11013)]. Here it is shown (and this extends previously obtained results) that the equation of the title implies that \(k\) is bounded, and for fixed \(k\) precise conditions are given under which the equation implies that both \(x\) and \(y\) are bounded by an effectively computable constant depending only on \(m\), \(d_ 1\), and \(d_ 2\).