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I am interested in learning a bit more about symmetric functions of $n$ variables, namely functions that are invariant under permutation of their arguments : $\forall \pi \in \sigma_n$, $$f(x_1,...,x_n) = f(x_{\pi(1)},...,x_{\pi(n)})$$

When $f$ is a polynomial or a rationial function, it can be written in a unique way as a sum of elementary symmetric polynomials or rational functions. This is the fundamental theorem of symmetrical functions (see http://mathworld.wolfram.com/FundamentalTheoremofSymmetricFunctions.html for instance).

What can be said about a symmetric function that is not a polynomial or a rational function (only $C^\infty$ or analytic for instance) ?

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    I think in the case of analytic functions that if you represent the coefficients as an $n$ dimensional array, it will be symmetric. – 2017-02-23
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    Okay. I was more hoping for something resembling the fundamental theorem of symmetric functions. It would probably look like "there exists a unique analytic (or $C^\infty$) functions such that $ f(x_1,...,x_n) = g(e_1,...,e_n)$ where the $e_i$ are elementary symmetric polynomials". But I'm looking for a more solid evidence than just intuition. – 2017-02-23
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    The formulation of the fundamental theorem of symmetric function in the question is extremely sloppy. It is not a sum of such and such, but a polynomial/rational function in terms of those (elementary symmetric functions). – 2018-07-10

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