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 $C_n$. In a special $n=1$ case, the simplest $p\to 0$ 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 $C_n$ and $A_n$ root systems and corresponding symmetry transformations are considered. An application of the rarefied type II $C_n$ elliptic hypergeometric function to some eigenvalue problems is briefly discussed.