Under suitable asymptotic and convexity conditions on a function $g\colon\mathbb{R}_+\to\mathbb{R}$, the solution to $Δf=g$, where $Δ$ is the forward difference operator, is unique up to an additive constant and is called the principal indefinite sum of $g$, generalizing the additive form of Bohr-Mollerup's theorem. We consider the map $Σ$, which assigns to each admissible function $g$ its principal indefinite sum that vanishes at $1$, and we naturally explore its iterates, which produce repeated principal indefinite sums, in analogy with the concept of repeated indefinite integrals. Explicit formulas and convergence results are established, highlighting connections with classical combinatorial and special functions, including the multiple gamma functions, for which we also provide integral representations.

翻译：在函数 $g\colon\mathbb{R}_+\to\mathbb{R}$ 满足适当的渐近性和凸性条件下，方程 $Δf=g$（其中 $Δ$ 为前向差分算子）的解在相差一个加性常数的意义下是唯一的，并称为 $g$ 的主不定求和，这推广了 Bohr-Mollerup 定理的加性形式。我们考虑映射 $Σ$，它将每个容许函数 $g$ 映射到其在 $1$ 处为零的主不定求和，并自然地探讨其迭代，从而产生重复主不定求和，类比于重复不定积分的概念。我们建立了显式公式和收敛性结果，突出了其与经典组合函数及特殊函数（包括多重伽马函数，我们也为其提供了积分表示）的联系。