# 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