We investigate Diophantine problems concerning linear combinations of polynomials of the shape $$a_0 x+a_1x(x+1)+ a_2x(x+1)(x+2)+ \cdots + a_n x(x+1)\ldots (x+n)$$ a 0 x + a 1 x ( x + 1 ) + a 2 x ( x + 1 ) ( x + 2 ) + ⋯ + a n x ( x + 1 ) … ( x + n ) with $$n\in \mathbb{N}\cup\{0\}$$ n ∈ N ∪ { 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 Bérczes, Hajdu, Luca and the author [2] given for the sums $$x+x(x+1)+x(x+1)(x+2)+ \cdots + x(x+1)\ldots (x+n)$$ x + x ( x + 1 ) + x ( x + 1 ) ( x + 2 ) + ⋯ + x ( x + 1 ) … ( x + n ) , i.e., for the case $$a_0=a_1= \cdots = a_n = 1$$ a 0 = a 1 = ⋯ = a n = 1 . Our work is closely connected also with some results of Tengely and Ulas [15] concerning the case when the coefficients $$a_0,a_1, \ldots, a_n$$ a 0 , a 1 , … , a n are zeroes and ones.

中文翻译：

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关于连续整数乘积的线性组合

我们研究关于形状 $$a_0 x+a_1x(x+1)+ a_2x(x+1)(x+2)+ \cdots + a_n x(x+1)\ldots (x +n)$$ a 0 x + a 1 x ( x + 1 ) + a 2 x ( x + 1 ) ( x + 2 ) + ⋯ + anx ( x + 1 ) … ( x + n ) 与 $$n \in \mathbb{N}\cup\{0\}$$ n ∈ N ∪ { 0 } 。我们为这些线性组合的幂、位移幂和二次多项式值提供了有效的有限性结果，概括了 Hajdu、Laishram 和 Tengely [10] 以及 Bérczes、Hajdu、Luca 和作者 [2] 给出的类似结果总和 $$x+x(x+1)+x(x+1)(x+2)+ \cdots + x(x+1)\ldots (x+n)$$ x + x ( x + 1 ) + x ( x + 1 ) ( x + 2 ) + ⋯ + x ( x + 1 ) … ( x + n ) ，即对于 $$a_0=a_1= \cdots = a_n = 1$$ a 0 = a 1 = ⋯ = an = 1 。