I know that the Abell-Ruffini theorem prove that we cannot solve a general equation of degree $n>4$ with radicals. But I've read that quintic equations can be solved by means of elliptic modular functions or generalized hypergeometric functions and this can be done also for other equations of higher degree.
I intend that to find the solutions of a high degree equation numerical methods are more efficient than analytical methods, but I'm curious to know if there is some general method that, using some kind of special functions, can be applied to solve equations of any degree. Or any equation of different degree require different special functions? And, there is some method to identifiy the equations of degree greater than four that can be solved by radicals?
References to accessible works about this topic are wellcome.