An Application of Okada's Minor Summation Formula
Combinatorics
2007-05-23 v1
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 is the set of -tuples satisfying . 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.
Cite
@article{arxiv.math/9805108,
title = {An Application of Okada's Minor Summation Formula},
author = {David P. Robbins},
journal= {arXiv preprint arXiv:math/9805108},
year = {2007}
}