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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.

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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).

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  • $\begingroup$ That's pretty good, but I must remind the reader that the sextic equations can only be reduced to 2-parameter sextic equation $x^6 + a x^2 + b x + b = 0$ by the usual method... $\endgroup$
    – Ember Edison
    Commented May 10, 2022 at 12:02
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    $\begingroup$ In fact any trinomial sextic having this form $z^6-\beta z^m+\gamma=0$, for $m\in\{0,1,..,5\}$ can be put in terms of Fox-Wright $_2\Psi_1$ function that is a special case of FoxH. For $m$ integer it can be also set in terms of a series of generalized hypergeometric functions. $\endgroup$
    – Jorge Zuniga
    Commented May 10, 2022 at 13:44

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