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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.)