A short proof of a Landen product identity for the Grötzsch ring function
arXiv:1509.01915
Abstract
We give a short, self-contained proof of a closed form for the infinite product associated with the Landen transformation of the Grötzsch ring function, combining the arithmetic-geometric mean, theta duplication identities, and an elementary coefficient comparison. The identity itself is the equality case of a 1999 product theorem of Qiu and Vuorinen, and we record what is known and what apparently cannot exist about its extensions.
arXiv comment5 pages