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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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    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.2013-02-10

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