Sums of powers of roots via Bell polynomials

Abstract

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.

Keywords:

• Bell polynomials
• generating functions
• Girard's formula

2010 Mathematics Subject Classification:

• 33E99

Acknowledgments

This note was inspired by reading [2]. The authors would like to thank the Editor and the referee for careful reading and comments which greatly improved the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).