Rarefied elliptic hypergeometric functions

Two exact evaluation formulae for multiple rarefied elliptic beta integrals related to the simplest lens space are proved. They generalize evaluations of the type I and II elliptic beta integrals attached to the root system Cn. In a special n=1 case, the simplest p→0 limit is shown to lead to a new class of q-hypergeometric identities. Symmetries of a rarefied elliptic analogue of the Euler-Gauss hypergeometric function are described and the respective generalization of the hypergeometric equation is constructed. Some extensions of the latter function to Cn and An root systems and corresponding symmetry transformations are considered. An application of the rarefied type II Cn elliptic hypergeometric function to some eigenvalue problems is briefly discussed.