# How to solve the general sextic equation with Kampé de Fériet functions?

It is frequently stated, for example on Wolfram Mathworld, that the general sextic equation

$$x^6 + a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x^1 + a_0 = 0$$

can be solved in terms of Kampé de Fériet functions - however the explicit solution is hard to find.

How can one solve the general sextic equation with Kampé de Fériet functions (or indeed other functions)?

• I hope this may help math.stackexchange.com/questions/1313118/… Commented Jul 23, 2016 at 18:54
• The general sextic first has to be reduced to the form $x^6+x^2+ax+b = 0$, then one solves it by the two-variable Kampé de Fériet functions. But I've never seen a specific example in all this time. Commented Jun 23, 2023 at 17:31

Just in brief (I need to go catch a train), here are the links to the papers cited on the MathWorld page. They are all open domain (published more than a century ago!)

Here is the link for the first paper cited on the MathWorld page

Similarly here is the second paper by Coble on the subject.

And finally, the last paper Weisstein cites.

This is a paper on solution methods for sextics in general.

And if any of the links fails, a simple Google search should reveal the original paper.

I am not sure if an explicit solution is to be found in any of these papers, but if one cannot be deduced, then the Wolfram MathWorld page on the subject has a problem of its own!