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Repeated principal indefinite summation

Thomas Lamby, Jean-Luc Marichal

Abstract

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.

Repeated principal indefinite summation

Abstract

Under suitable asymptotic and convexity conditions on a function , the solution to , where is the forward difference operator, is unique up to an additive constant and is called the principal indefinite sum of , generalizing the additive form of Bohr-Mollerup's theorem. We consider the map , which assigns to each admissible function its principal indefinite sum that vanishes at , 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.
Paper Structure (11 sections, 22 theorems, 139 equations, 1 figure)

This paper contains 11 sections, 22 theorems, 139 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Iterating the Analog of Gauss' Limit
  4. A Characterization
  5. A Generalization of Malmstén's Formula
  6. Further Examples
  7. Polynomial functions
  8. The Reciprocal Function
  9. Powers of the Reciprocal Function
  10. Negative Falling Factorials
  11. Discrete Analog of Cauchy's Formula for Repeated Integration

Key Result

Proposition 2.2

Let $I$ be an open real interval, let $f\colon I\to\mathbb{R}$ be a function, and let $p\in\mathbb{N}$. Then the following assertions hold:

Figures (1)

  • Figure 1: Graph of $g(x)$ on $(0,5]$

Theorems & Definitions (50)

  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6: Multiple Gamma Functions
  • Proposition 2.7
  • proof
  • ...and 40 more