Rarefied elliptic hypergeometric functions

TL;DR

This paper introduces new evaluations and identities for rarefied elliptic hypergeometric functions, extending classical hypergeometric functions and exploring their symmetries, with applications to eigenvalue problems.

Contribution

It provides exact evaluation formulas for rarefied elliptic beta integrals, generalizes hypergeometric functions to root systems, and constructs related differential equations.

Findings

Derived new evaluation formulas for rarefied elliptic beta integrals.

Established symmetries and generalizations of elliptic hypergeometric functions.

Connected these functions to eigenvalue problems in mathematical physics.

Abstract

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.