5
$\begingroup$

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.

$\endgroup$
0

1 Answer 1

7
$\begingroup$

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).

$\endgroup$
2
  • $\begingroup$ 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... $\endgroup$ May 10, 2022 at 12:02
  • 2
    $\begingroup$ 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. $\endgroup$ May 10, 2022 at 13:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.