ABSTRACT

Let $p_1,p_2,\dots ,p_n$ be distinct positive real numbers and m be any integer. Every symmetric polynomial $f(x,y)\in C[x,y]$ induces a symmetric matrix ${\left[f(p_i,p_j)\right]}_{i,j=1}^n.$ We obtain the determinants of such matrices with an aim to find the determinants of $P_m={\left[(p_i+p_j{)}^m\right]}_{i,j=1}^n$ and $B_{2m}={\left[(p_i-p_j{)}^{2m}\right]}_{i,j=1}^n$ for $m\in \mathbb{N}$ (where $\mathbb{N}$ is the set of natural numbers) in terms of the Schur polynomials. We also discuss and compute determinant of the matrix $K_m={\left[\frac{p_i^m+p_j^m}{p_i+p_j}\right]}_{i,j=1}^n$ for any integer m in terms of the Schur and skew-Schur polynomials.

中文翻译：

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一些特殊矩阵的行列式

摘要

让$p_1,p_2,\dots ,p_n$是不同的正实数，m是任意整数。每个对称多项式$F(X,\mathrm{是的})\in C[X,\mathrm{是的}]$引出一个对称矩阵${\left[F(p_{\mathrm{一世}},p_j)\right]}_{\mathrm{一世},j=1}^n.$我们获得这些矩阵的行列式，目的是找到${磷}_{米}={\left[(p_{\mathrm{一世}}+p_j{)}^{米}\right]}_{\mathrm{一世},j=1}^n$和${乙}_{2米}={\left[(p_{\mathrm{一世}}-p_j{)}^{2米}\right]}_{\mathrm{一世},j=1}^n$为了$米\in \mathbb{ñ}$（在哪里$\mathbb{ñ}$是自然数的集合）根据 Schur 多项式。我们还讨论和计算矩阵的行列式${ķ}_{米}={\left[\frac{p_{\mathrm{一世}}^{米}+p_j^{米}}{p_{\mathrm{一世}}+p_j}\right]}_{\mathrm{一世},j=1}^n$对于Schur 和 skew-Schur 多项式中的任何整数m 。