What is closed-form expression for the summation
$$ S(n,m)=\sum_{|\alpha|=m} p^{\alpha} = \sum_{\alpha_1 + \cdots + \alpha_n = m} \prod_{i=1}^n p_i^{\alpha_i} $$ as a function of $n$ and $m$? Here $\alpha=(\alpha_1,\cdots,\alpha_n)$ is a $n$-tuple of non-negative integers.
For specific values of $n$ the sum has a closed form, e.g. $S(1,m)=p_1^m, S(2,m)=\frac{p_2^{m+1}-p_1^{m+1}}{p_2-p_1}$, etc. I wonder if there is a more manageable form with few summations, even though there might be no closed form for the general expression.
I think that $$S(3,m)=\frac{p_1^{m+2}}{(p_1-p_2)(p_1-p_3)}+\frac{p_2^{m+2}}{(p_2-p_1)(p_2-p_3)}+\frac{p_3^{m+2}}{(p_3-p_1)(p_3-p_2)}$$ which is promising.
$$S(k,m)=\sum_{i=1}^k\frac{p_i^{m+k-1}}{\prod_{j=1,j\neq i}^k(p_i-p_j)}$$
To get that, use the recursion $$S(3,m)=S(2,m)+p_3S(2,m-1)+p_3^2S(2,m-2)+,,,$$ together with your formula for $S(2,m)$.