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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.)



Reference request on symmetric polynomials
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. 


