On Bottcher's mysterious identity

In his work on Wiener-Hopf determinants, A. Böttcher came across what he termed a “mysterious” identity that evaluates a certain sum of a rational function of a primitive root of unity in terms of the Barnes Gfunction, which was later generalized by Basor and Forrester. We give a direct proof of Böttcher’s identity and its many generalizations by using elements of the theory of symmetric functions.