Consider a function with a power series expansion and roots $\{x_n,1\le n<\mathrm{\infty }\}$. We give two methods for obtaining the sums of powers of these roots, $S_k=\sum _{n=1}^{\mathrm{\infty }}{x}_n^k.$The first is by expressing Girard's formula in terms of Bell polynomials. Explicit expressions are given for $S_k$ for $k\le 10$. The second method is through generating functions.

中文翻译：

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通过贝尔多项式求根的幂和

考虑一个具有幂级数展开和根的函数$\{X_n,1\le n<\mathrm{\infty }\}$. 我们给出了两种方法来获得这些根的幂和，${\mathrm{小号}}_{ķ}=\sum _{n=1}^{\mathrm{\infty }}{X}_n^{ķ}.$第一种是用贝尔多项式表达吉拉德公式。给出了明确的表达式${\mathrm{小号}}_{ķ}$为了$ķ\le 10$. 第二种方法是通过生成函数。