TITLE: Is it possible to solve sextic equations using the Fox H function?
QUESTION [5 upvotes]: Although the Kampé de Fériet function can solve the sextic equation, the details about it are shrouded in the fog of more than a century ago.
In contrast, we know more about the Fox H function, and we have even succeeded in implementing it on CAS.
I would like to know if the experts of the Fox H function have a more modern understanding of the solution of the sextic equation.

REPLY [6 votes]: Trinomial sextic equations $z^6-z-t=0$ can be solved in terms of the Fox H function:
$$z_j=\tfrac{1}{5}t H_{2,3}^{1,2}\biggl(te^{j\frac{2 i \pi }{5}}\biggl|
\begin{array}{c}
(0,1),(0,6/5)  \\
 (0,1),(-1,1),(0,1/5) \\
\end{array}
\biggr)+e^{-j\frac{2 i \pi }{5}},\;\;j\in\{0,1,2,3,4,5\}.$$
Mathematica input:
(t/5)*FoxH[{{{0,1},{0,6/5}},{}},{{{0,1}},{{-1,1},{0,1/5}}},t Exp[(2 Pi I)/5]^j]+Exp[(2 Pi I)/5]^-j

This is a special case of a more general solution of trinomial equations of arbitrary order, $z^\alpha-\beta z+\gamma=0$, described in All the trinomial roots, their powers and logarithms from the Lambert series, Bell polynomials and Fox–Wright function (J. Math. Chem. 57 (2019) pp 59–106).