Product as the sum of powers times a symmetric polynomial: What's the name of this property and what is it used for?

Question

I noticed that the product of a group of positive integers $N$ with $n$ elements can be expressed as the sum of powers of the smallest member of $N$ times some (what I later found out be called) elementary symmetric polynomial. Suppose $N$ is sorted in increasing order such that $N_1$ is the smallest member of the set. Let $D$ be the set of $j$ elements, containing the values of $N_i-N_1$ that are $> 0$ (ex: $N=\{2,2,5,5,6\} \mapsto D=\{5-2=3, 5-2=3, 6-2=4\}$).Let $SPD_i$ denote the $i$th elementary symmetric polynomial on the elements of $D$. Then: $$\prod_{k = 1}^{n} N_k = \sum_{i = 0}^{j} SPD_i N_1^{n-i}$$ Using the sample set I gave above ($n=5, N_1=2, D=\{3,3,4\})$: $$\prod_{k = 1}^{n} N_k = 2*2*5*5*6 = 600$$ $$1*2^5+(3+3+4)*2^4+(3*3+3*4+3*4)*2^3+(3*3*4)*2^2=600$$ I didn't know anything about symmetric polynomials at the time I noticed it, and honestly I still don't know much. Although it seems there's plenty of material about symmetric polynomials online, I couldn't find anything about this specific application of it that I've just described. Does it have a name? As it's fairly general, I thought it would have at least a couple of applications but found none. Is it useful at all?

Answer 1

Edit: fixed subscripts and reformulated. Let $m$ be the largest index for which $N_m=N_1$. Consider $$P(x)=\prod_{k=m+1}^n(x+N_k-N_1).$$ Then in terms of elementary symmetric polynomials $$P(x) = \sum_{k=m}^n e_{n-k}(N_{m+1}-N_1, \ldots, N_n-N_1) x^{k-m}$$ (with the convention that $e_0=1$). Substitute $x\leftarrow N_1$ to get $$P(N_1)=\prod_{k=m+1}^nN_k = \sum_{k=m}^n e_{n-k}(N_{m+1}-N_1, \ldots, N_n-N_1) N_1^{k-m}.$$