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I asked this question on MO, but it has been tagged off-topic.

Is there any analogue of the method for expressing roots of polynomials of degree $5$ with elliptic and $\eta$-functions that generalizes to polynomials of degrees $n>5$?

More specifically: does there exist any ''reasonable'' sequence of ''reasonable'' finite sets of ''reasonable'' special functions such that for arbitrary polynomial of degree $d$ its roots are expressible by a unique term made out of coefficients of polynomial and functions from the $d$-th set?

By ''reasonable'' special functions I mean ones that has been investigated independently or have been already coined and are decently understood or are solutions to reasonable ODEs. By a reasonable sequence I mean basically something like a family of functions parametrized by $n \in \mathbb{N}.$

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  • $\begingroup$ I present a bit of this material in a wider context in this Computational Science answer, with some links for fuller exposition. $\endgroup$ – hardmath Sep 5 '14 at 18:48

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