By means of matrix decompositions, three determinants with their entries being binomial sums are evaluated in closed forms. Ten remarkable examples are illustrated as propositions, which present determinant identities about binomial coefficients and quotients of rising factorials, as well as orthogonal polynomials named after Hermite, Laguerre and Legendre.

中文翻译：

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三个行列式评估

通过矩阵分解，以封闭形式评估三个行列式，其项为二项式和。命题举例说明了十个杰出的例子，它们给出了关于二项式系数和上升阶乘的商的决定性恒等式，以及以Hermite，Laguerre和Legendre命名的正交多项式。