Eric Rains

Abstract:   Euler's beta (and gamma) integral lies at the core of much of the theory of special functions, and many generalizations have been studied, including multivariate analogues (the Selberg integral; also work of Dixon and Varchenko), $q$-analogues (Askey--Wilson, Nasrallah--Rahman), and both (work of Milne--Lilly and Gustafson). In 2001, van Diejen and Spiridonov conjectured several generalizations going beyond $q$ to the elliptic level (replacing $q$ by a point on an elliptic curve). I'll discuss how a simple question in random matrix theory led to a proof of their conjectured identities, in fact generalizing them to a transformation a la the integral representation of hypergeometric functions). I'll also discuss an elliptic Selberg integral with a (partial) symmetry under the Weyl group $E_8$, as well as connections with the theory of Macdonald and Koornwinder polynomials.