Okada次求和公式的一个应用
An Application of Okada's Minor Summation Formula
摘要：
Noam Elkies和Everett Howe分别注意到了多重积分\int_R \prod_{1 \le i < j \le k}（x_j-x_i）dx_1 \cdots dx_k的一个优美的乘积公式，其中区域$R$是满足$0 < x_1 < \cdots < x_k < 1$的$k$元组的集合。后来这个公式原来是一个特殊情况下的一个公式塞尔伯格。我们证明了一个明显不同的推广\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）}.关键的工具是一个显着的身份冈田求和的k乘k未成年人的n乘k矩阵的限制形式。
英文摘要：
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.
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