Let $m_1\ge m_2\ge\ldots\ge m_k$ are nonnegative integers. Then, we can consider the following product of power sums:
$$
p_m(x_1,\ldots,x_n)=\prod_{i=1}^{k}\sum_{j=1}^{n}x_j^{m_i}.
$$
Since Schur polynomials $\{s_{\lambda}\}$ ($\lambda$ is a partition with $n$ nonnegative parts) form a basis in the space of the symmetric polynomials in $n$ variables $x_1,\ldots,x_n$, we can represent $p_m$ as a linear combination of $s_{\lambda}$.
Question 1: Does there exist a formula for the corresponding coefficients? I almost sure that there is such formula in Hall's book but I haven't found it yet.
Question 2: Even if this formula exist, I suppose that they consider only nonnegative $m_i$. I am also interested in the case when some of $m_i$ are negative. However, I don't know whether there exist a similar formula. (We can also define Schur functions for $m=(m_1,\ldots,m_n)$ via bialternant formula, cf. Weyl character fromula)
Some motivation: it's known that Schur functions are characters of irreducible (unitary) representations of the $U(n)$, so we encounter the following problem: decide whether a given function is a character of some representation of the $U(n)$ or not. In other words, we need to check whether this function can be represented as sum of Schur functions with nonnegative coefficients or not. That's why I am interested in exact formula for coefficients for $p_m$.
Remark. For simplicity we can suppose that number $n$ of variables is large enough (or work in the ring of the symmetric functions).
Best Answer
Partial answer. I found the answer for the first question (cf. Symmetric functions). It turns out that in the ring of symmetric functions the following identity holds: $$ p_{\mu}(x)=\sum_{\lambda}\chi_{\mu}^{\lambda}\cdot s_{\lambda}(x), $$ where $\chi_{\mu}^{\lambda}$ is a value of the character $\chi^{\lambda}$ of the irreducible representation of $S_n$ which corresponds to partition $\lambda\vdash n$ (or, equivalently, Young diagram) on an element $\sigma\in S_n$ whose cycle structure corresponds to the partition $\mu\vdash n$.
Moreover, there is a formula for $\chi_{\mu}^{\lambda}$ (the so called Murnaghan-Nakayama rule) and the positivity question can be resolved via orthogonal relations between characters of the irreducible representations.
However, the question is still open if we allow "partitions" with negative "parts" (see question for more details).