On gamma quotients and infinite products

On gamma quotients and infinite products
Marc Chamberland, Armin Straub — Advances in Applied Mathematics — Volume 51, Number 5, 2013, Pages 546-562


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.

Corrigendum: In Example 2.4, "z" needs to be replaced by "-z" on the right-hand side of the first equation.


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    author = {Marc Chamberland and Armin Straub},
    title = {On gamma quotients and infinite products},
    journal = {Advances in Applied Mathematics},
    year = {2013},
    volume = {51},
    number = {5},
    pages = {546--562},
    doi = {10.1016/j.aam.2013.07.003},