On gamma quotients and infinite products

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 Ramanujanʼs 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.