I want to write the function $$ F_N=\sum_{i=1}^N\sqrt{x_i} $$ in terms of the $N$ elementary symmetric polynomials of the $N$ positive variables $x_1,\dots,x_N$.
The $N=1$ case is trivial, as we have $F_1=\sqrt{x_1}$ so $$F_1=\sqrt{s_1}$$
The $N=2$ case is simple, as we have $F_2^2=(\sqrt{x_1}+\sqrt{x_2})^2=x_1+x_2+2\sqrt{x_1x_2}$, which means that $$F_2=\sqrt{s_1+2\sqrt{s_2}}$$
The $N=3$ case is not so simple, but after a bit one finds $$F_3=\sqrt{s_1+2\sqrt{s_2+2\sqrt{s_3}F_3}}$$ so one can at least solve for $F_3$.
There is kind of a pattern evolving here. However, I cannot proceed beyond $N=3$... Any insights?
EDIT: I'm not sure this is helpful, but as $F_N$ is an eigenfunction of the operator $2\sum_ix_i\frac{\partial}{\partial x_i}$, one can easily find that $$F_N=\sqrt{s_1}G_N\left(\frac{s_2}{s_1^2},\frac{s_3}{s_1^3},\dots,\frac{s_N}{s_1^N}\right)$$ where $G_N$ is a function of $N-1$ arguments.
EDIT 2: Actually, $F_N$ is an eigenfunction of the operators $$\frac{2 (-1)^{n-1}}{(1/2)_{n-1}}\sum_ix_i^n\frac{\partial^n}{\partial x_i^n}$$ for every positive integer $n$, where $(a)_n$ is the Pochhammer symbol. I wonder if one can use this fact to further reduce the structure of $F_N$.