An Application of Okada's Minor Summation Formula

arXivmath/9805108MaRDI QIDQ6500910

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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 detleft(x_i^{a_j-1} ight)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.

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