An Application of Okada's Minor Summation Formula

Abstract

Noam Elkies and Everett Howe independently noticed a certain elegant product formula for the multiple integral \int_R \prod_{1 \le i < j \le k} (x_j-x_i) dx_1 \cdots dx_k, where the region $R$ is the set of $k$-tuples satisfying $0 < x_1 < \cdots < x_k < 1$. Later this formula turned out to be a special case of a formula of Selberg. We prove an apparently different generalization \int_R \det\left(x_i^{a_j-1}\right)dx_1 \cdots dx_k = {\prod_{1 \le i<j \le k}(a_j-a_i)\over \prod_{1 \le i \le k} a_i \prod_{1 \le i<j \le k} (a_j+a_i)}. The key tool is a limiting form of a remarkable identity of Okada for summing the k by k minors of an n by k matrix.