Multiple sine, multiple elliptic gamma functions and rational cones
Abstract
We define generalizations of the multiple elliptic gamma functions and the multiple sine functions, labelled by rational cones in
R
r
. For
r=2,3
we prove that the generalized multiple elliptic gamma functions enjoy a modular property determined by the cone. This generalizes the modular properties of the elliptic gamma function studied by Felder and Varchenko. The generalized multiple sine enjoy a related infinite product representation, generalizing the results of Narukawa for the ordinary multiple sine functions.