关于Mina矩阵及其相关行列式恒等式的一些注释

Some Notices on Mina Matrix and Allied Determinant Identities

中文关键词:  Mina行列式恒等式  Mina矩阵  LU分解  求导算子  反演  组合恒等式  Lagrange插值  Melzak公式

英文关键词:Mina determinant identity  Mina matrix  LU factorization  derivative operator  inverse  combinatorial identity  Lagrange interpolation  Melzak's formula

中文摘要:

      本文通过对所谓的Mina矩阵$\mathbf{D}^{n}_x(f^{a_k}(x))$建立的LU分解,不仅得到Mina行列式恒等式的一个初等证明,而且还给出了Mina矩阵的逆矩阵. 进一步地,通过建立在拉格朗日插值公式上的矩阵分解,本文给出了Mina型矩阵的两个新的行列式恒等式.

英文摘要:

      By means of a special LU factorization of the Mina matrix with the $n$-th row and $k$-th column entry $\mathbf{D}^{n}_x(f^{a_k}(x))$, we obtain not only a short proof of the Mina determinant identity but also the inverse of the Mina matrix. Finally, by use of some similar factorizations built on the Lagrange interpolation formula, two new determinant identities of Mina type are established.