Repeated principal indefinite summation

TL;DR

This paper generalizes the concept of indefinite summation through repeated principal indefinite sums, providing explicit formulas, convergence results, and connections to classical functions like multiple gamma functions.

Contribution

It introduces the notion of repeated principal indefinite sums, extending classical summation concepts and linking them to special functions such as multiple gamma functions.

Findings

Explicit formulas for repeated principal indefinite sums

Convergence results for the iterates of the summation map

Connections established with classical combinatorial and special functions

Abstract

Under suitable asymptotic and convexity conditions on a function $g\colon\mathbb{R}_+\to\mathbb{R}$, the solution to $\Delta f=g$, where $\Delta$ 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 $\Sigma$, 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.