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Assume that I have an analytic, symmetric function $F(x,y,z)$. Such a function should have an expansion of the form $$ F(x,y,z) = \sum_{k,m,n\geq 0} c_{k,m,n} e_1^ne_2^m e_3^k$$
where $e_1,e_2,e_3$ are the elementray symmetric polynomials. How would I go about calculating these $c_{m,n,k}$ in an efficient way (by efficient I mean something better than rewriting the polynomials $$ P_{n_1,n_2,n_3}(x_1,x_2,x_3) = \sum_{\sigma \in S_3} x^{n_{\sigma(1)}}y^{n_{\sigma(2)}}z^{n_{\sigma(3)}}$$ in terms of elementary polynomials by hand...).

More concretely I would be interested in knowing whether the function $$ G(x_1,x_2,x_3) = \sum_{\sigma \in S_3} \operatorname{exp}\Big[ Ax_{\sigma(1)}+Bx_{\sigma(2)}+Cx_{\sigma(3)}\Big] $$ can be rewritten as a function in elementary symmetric polynomials in a nice form.

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I'm not entirely convinced that this is possible. Consider $\sum(x^n+y^n+z^n)$. Certainly for each $n$ we can write $x^n+y^n+z^n$ as a polynomial in $x+y+z,xy+xz+yz,xyz$ but maybe the coefficient of, say, $xyz$ doesn't go to zero fast enough for the sum over $n$ to converge. Is there a theorem guaranteeing the existence of the kind of expansion you want? –  Gerry Myerson Feb 11 '12 at 23:51
    
Using the geometric series I think your function can be rewritten as a rational function in $e_1,e_2$ and $e_3$ - which could then be expanded into a power series. In general I would say that writing my function $F$ in terms of elementary functions using term-wise approach, would me provide with a new series which for $e_1,e_2,e_3$ small enough will converge since the original series did (writing out $e_i$ as polynomials in $x,y,z$ should give my original series back by uniqueness of the Taylor expansion). –  newguy Feb 12 '12 at 0:08
    
You can express this as an expansion into orthogonal functions (à la Fourier) by taking the correct base. I don't know if that is possible, or if the orthogonal base turns out in any way pleasant, but it might be worth a try. –  vonbrand Feb 10 '13 at 2:22
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