We start from an interpretation of the BC2-symmetric “Type I” (elliptic Dixon) elliptic hypergeometric integral evaluation as a formula for a Casoratian of the elliptic hypergeometric equation, and give an extension to higher-dimensional integrals and higher-order hypergeometric functions. This allows us to prove the corresponding elliptic beta integral and transformation formula in a new way, ...

Euler’s work on elliptic integrals is a milestone in the history of algebraic geometry. The founders of calculus understood that some algebraic functions could be integrated using elementary functions (logarithms and inverse trigonometric functions). Euler realized that integrating other algebraic functions leads to genuinely different functions, elliptic integrals. These functions are not some...

We give a brief review of the main results of the theory of elliptic hypergeometric functions — a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler’s beta integral, which is called the elliptic beta integral. An elliptic analogue of the Gauss hypergeometric function is constructed together with the elliptic ...