# The Roman harmonic numbers revisited

## TL;DR

This paper revisits Roman harmonic numbers, exploring their properties, deriving new relations, and demonstrating their connection to derivatives of Pochhammer symbols, thereby revitalizing their mathematical significance.

## Contribution

It provides a comprehensive review of Roman harmonic numbers, introduces new properties, and links them to derivatives of special functions, enhancing their theoretical understanding.

## Key findings

- Derived integral representation and generating relations for Roman harmonic numbers.
- Established sum rules involving these numbers.
- Connected Roman harmonic numbers to derivatives of Pochhammer symbols.

## Abstract

Two decades ago, Steven Roman, Daniel E. Loeb and Gian-Carlo Rota introduced a family of harmonic numbers in their study of harmonic logarithms. We propose to refer to those numbers as {\it Roman harmonic numbers}. With the purpose of revitalizing the study of these mathematical objects, we recall here their known properties and unveil additional ones. An integral representation, several generating relations, and a collection of sum rules involving those numbers are presented. It is also shown that higher derivatives of the Pochhammer and reciprocal Pochhammer symbols are easily expressed in terms of Roman harmonic numbers.