Determinantal elliptic Selberg integrals

Abstract

The classical Selberg integral contains a power of the Vandermonde determinant. When that power is a square, it is easy to prove Selberg's identity by interpreting it as a determinant of one-variable integrals. We give similar proofs of summation and transformation formulas for continuous and discrete elliptic Selberg integrals. In the continuous case, the same proof was previously given by Noumi. Special cases of these identities have found applications in combinatorics.