On linear combinations of products of consecutive integers

Abstract

We investigate Diophantine problems concerning linear combinations of polynomials of the shape $$a_0 x+a_1x(x+1)+ a_2x(x+1)(x+2)+ \cdots + a_n x(x+1)\ldots (x+n)$$ with $$n\in \mathbb{N}\cup\{0\}$$ . We provide effective finiteness results for the power, shifted power, and quadratic polynomial values of these linear combinations, generalizing the analogous results of Hajdu, Laishram and Tengely [10], and of Berczes, Hajdu, Luca and the author [2] given for the sums $$x+x(x+1)+x(x+1)(x+2)+ \cdots + x(x+1)\ldots (x+n)$$ , i.e., for the case $$a_0=a_1= \cdots = a_n = 1$$ . Our work is closely connected also with some results of Tengely and Ulas [15] concerning the case when the coefficients $$a_0,a_1, \ldots, a_n$$ are zeroes and ones.