Let $\mathcal{S}_{x}=\{x_{1,},x_{2},\ldots x_{n}\}$ be a set of $n$ indeterminates. The $h^{th}$elementary symmetric polynomial is the sum of all monomials with $h$ factors \begin{eqnarray*} e_{h}(\mathcal{S}_{x}) & = & \sum_{1\leqslant i_{1}<i_{2}<\ldots<i_{h}\leqslant n}x_{i_{1}}x_{i_{2}}\ldots x_{i_{h-1}}x_{i_{h}} \end{eqnarray*} which, from a generating function standpoint, can be built up as the coefficients of the $h^{th}$ power of the following linear factorization \begin{eqnarray*} \prod_{i=1}^{n}(1+x_{i}z) & = & (1+x_{1}z)(1+x_{2}z)(1+x_{3}z)\ldots(1+x_{n}z)\\ & = & \sum_{h=0}^{n}e_{h}(\mathcal{S}_{x})z^{h} \end{eqnarray*}

Some usual specializations of the set $\mathcal{S}_{x}$ lead to known families of numbers and multiplicative identities: binomial coefficients for $x_{i}=1_{i}$, to $q$-binomial coefficients for $x_{i}=q^{i}$ and Stirling numbers of the first kind for $x_{i}=i$;

(i) For $\mathcal{S}_{1}=\{1_{1},1_{2},1_{3},\ldots,1_{n}\}$ \begin{eqnarray*} (1+z)^{n} & = & (1+1_{1}z)(1+1_{2}z)(1+1_{3}z)\ldots(1+1_{n}z)\\ & = & \sum_{h=0}^{n}{n \choose h}z^{h} \end{eqnarray*} binomial coefficients arise $e_{h}(\mathcal{S}_{1})={n \choose h}$

(ii) For $\mathcal{S}_{q^{i}}=\{q,q^{2},q^{3}\ldots,q^{n-1},q^{n}\}$ \begin{eqnarray*} \prod_{i=1}^{n}(1+q^{i}z) & = & (1+q^{1}z)(1+q^{2}z)(1+q^{3}z)\ldots(1+q^{(n-1)}z)\\ & = & \sum_{h=0}^{n}{n \choose h}_{q}q^{{h+1 \choose 2}}z^{h} \end{eqnarray*} we get the $q$-binomial coefficients (or Gaussian coefficients) $e_{h}(\mathcal{S}_{q^{i}})={n \choose h}_{q}q^{{h+1 \choose 2}}$

(iii) And for $\mathcal{S}_{i}=\{1,2,3,\ldots n-1\}$ \begin{eqnarray*} \prod_{i=1}^{n-1}(1+iz) & = & (1+1z)(1+2z)(1+3z)\ldots(1+(n-1)z)\\ & = & \sum_{h=0}^{n}\left[\begin{array}{c} n\\ n-h \end{array}\right]z^{h} \end{eqnarray*} Stirling numbers of the first kind arise $e_{h}(\mathcal{S}_{i})=\left[\begin{array}{c} n\\ n-h \end{array}\right]$

In this context, are there other specializations of the set $\mathcal{S}_{x}=\{x_{1,},x_{2},\ldots x_{n}\}$ which lead to other families of numbers or identities?