Reference request on symmetric polynomials A version of this question on stackexchange got a few comments from one person and no answers.
Let $e_k$ be the $k$th-degree elementary symmetric polynomial in variables $x_1,\ldots,x_n$ (so in particular $e_k=0$ if $k>n$).  I don't know much about these polynomials.
Consider the rational function
$$
\frac{e_1+e_3+e_5+\cdots}{e_0+e_2+e_4+\cdots}.\tag{1}
$$
The simplest case is with just two variables: $$x_1\circ x_2=\dfrac{x_1+x_2}{1+x_1 x_2}.\tag{2}$$


*

*Is it known, in the sense of being in the refereed literature or at least widespread folklore, that the binary operation defined by $(2)$ is associative?

*Is it known that $x_1\circ \cdots\circ x_n$ equals the expression in $(1)$ (the proof is trivial, but is the fact mentioned in the literature)?

*Is it known that $\pm1$ are absorbing elements for this operation, so that if just on of the $x$s in $(1)$ equals $1$ then the whole expression in $(1)$, depending on $x_1,\ldots,x_n$, equals $1$, and similarly $-1$?  (If $x_1=1$ and $x_2=-1$, then you get $0/0$ and if I'm not mistaken, the singularity is not removable.)

 A: The fact that the binary operation defined by (2) is associative can be found in the Wikipedia article on formal groups (section 2) http://en.wikipedia.org/wiki/Formal_group. Presumably one might find more in the literature of formal groups. 
A: Not a complete answer, but note that 
\begin{align}
(*) \;\;\;e_k =e_k(x_1,\ldots,x_n)=\sum x_{i_1}x_{i_2}\cdots x_{i_k},
\end{align}
where the sum is over $1 \leq i_1 < i_2 < \ldots < i_k \leq n$. Moreover, for $1 \leq k< n$,
$$e_k|_{x_j=1}=e_k(x_1,\ldots,x_{j-1},x_{j+1},\ldots,x_n)+e_{k-1}(x_1,\ldots,x_{j-1},x_{j+1},\ldots,x_n),$$
and 
$$e_n|_{x_j=1}=e_{n-1}(x_1,\ldots,x_{j-1},x_{j+1},\ldots,x_n).$$
The $e_k$ term above comes from those summands in $(*)$ for which j does not occur among the $i_1,\ldots,i_k$, and the $e_{k-1}$ term comes from the remaining terms. So setting one of the $x_j=1$ in (1) above is clearly 1. (I'd have to think about plugging in $-1$ a little more).
I don't know off-hand where the best place is to look these particular things up, but Macdonald's "Symmetric Functions and Hall Polynomials" (the 2nd edition) would be a good start.
