Consider the system of equations \begin{align*} x_1^2 + x_2^2 + &\cdots + x_n^2 = q_2\\ x_1^3 + x_2^3 + &\cdots + x_n^3 = q_3\\ &\vdots\\ x_1^m + x_2^m + &\cdots + x_n^m = q_m \end{align*} where all the $q_i$ ($i = 2,\dots, m$) are known. Suppose that $m$ is arbitrary in the sense that for all $m > 1$, the value $q_m = x_1^m + x_2^m + \cdots + x_n^m$ is known (say, kept in a massive book somewhere). Can we determine $q_1 = x_1 + \cdots + x_n$?
Of course, we can, by taking $m = n$ and iteratively back-substituting the equations. However, I am wondering if there is a more clever way to go about this, that perhaps has a closed form.
A method I am more fond of is to relate the given system to that of a recursion. This works very well when we are given a lot of values of the Newton sums (in this case, in a massive book), or if we are given another variable in front of each of the $x_i$ terms (i.e. the system, $\sum_{i=1}^n a_i x_i^m=q_m$, for integers $m\geq 2$).
We can note that $q_m$ is in the form of an explicit formula for a recursion whose characteristic polynomial has roots $x_1,x_2,...x_n$.
This means we can express $q_m$ recursively as $q_m=\sum_{i=1}^n a_iq_{m-i}$, where $a_i$ is a sequence of length $n$. We can look up the values of $q_m$ from $q_2$ to $q_{2n+2}$ to get a system of $n$ linear equations and $n$ variables.
Once we solve for all the terms in $a_n$, we get that $q_1=\frac{q_{n+1}-\sum_{i=1}^{n-1} a_iq_{n+1-i}}{a_n}$