On gamma quotients and infinite products

TL;DR

This paper explores how infinite products involving rational functions can be expressed using gamma functions, demonstrating applications in number theory and proposing a numerical method for evaluating complex series.

Contribution

It shows the broad applicability of gamma function representations for infinite products and introduces a new numerical approach for slow-converging series.

Findings

Infinite products can be expressed via gamma functions.

Applications include multiplicative partitions and Ramanujan's notebooks.

A numerical method for slow series like Kepler--Bouwkamp constant.

Abstract

Convergent infinite products, indexed by all natural numbers, in which each factor is a rational function of the index, can always be evaluated in terms of finite products of gamma functions. This goes back to Euler. A purpose of this note is to demonstrate the usefulness of this fact through a number of diverse applications involving multiplicative partitions, entries in Ramanujan's notebooks, the Chowla--Selberg formula, and the Thue--Morse sequence. In addition, we propose a numerical method for efficiently evaluating more general infinite series such as the slowly convergent Kepler--Bouwkamp constant.