Repeated principal indefinite summation

Under suitable asymptotic and convexity conditions on a function $g\text{\hspace{0.17em}⁣}:{\mathbb{R}}_{+}\to \mathbb{R}$, the solution to $\mathrm{\Delta }f=g$, where $\mathrm{\Delta }$ 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 $\mathrm{\Sigma }$, 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.