Formula for certain specific polynomials


I have the following problem:

Given the polynomial $$ P_n (x_1, …, x_k) = x_1 ^ n + cdots + x_k ^ n $$ I would like to give an explicit formula (it depends on $ k, n $) to whom to write $ P_n (x_1, …, x_k) $ in terms of $ sigma_1 (x_1, …, x_k), …, sigma_n (x_1, …, x_k) $ (with $ sigma_i (x_1 ,. .., x_n) $ the $ i $elementary symmetric polynomial on $ x_1, …, x_n $), that is to say that I would like to find a formula for $ s_n $, or $$ P_n (x_1, …, x_k) = s_n ( sigma_1 (x_1, …, x_k), …, sigma_n (x_1, …, x_k)) $$

For example:

  • $ s_1 (x_1, …, x_k) = sigma_1 (x_1, …, x_n) $
  • $ s_2 (x_1, …, x_k) = sigma_1 ^ 2 (x_1, …, x_n) – 2 sigma_2 (x_1, …, x_n) $
  • $ s_3 (x_1, …, x_k) = sigma_1 ^ 3 (x_1, …, x_n) +3 sigma_3 (x_1, …, x_n) – 3 sigma_1 (x_1, …, x_n ) sigma_2 (x_1, …, x_n) $

Does anyone know how to attack this problem or have an idea?

I have already tried induction and recursion but I got nothing, I don't know if my calculations are wrong