I recently investigated $\textrm{Discriminant}_u(\omega P(u) +(z-u)P'(u))$, where $P(u) := u^3 + au + b$ and $\omega$ is a real parameter (with $\omega\in(0,1)\cup(1,3)$) associated with the order of a fractional derivative. (This is a special case of the problem previously discussed [here][1].) In doing so, I came across the following expression (plus some additional terms) that looks quite striking, but I haven't been able to place it:

$$\frac{3a(b-z^3)}{(\omega-1)\,\lambda} + \frac{z}{\lambda^3}\left(a^2\lambda\left(1-\frac{1}{(\omega-1)^2}\right)-3bz\right)$$

Here $\lambda:= \sqrt[3]{\omega-3}.$ I'm wondering if anyone has seen this expression, or something sufficiently similar to it, in any other context.

Due to a natural change of variables used to arrive at the above expression (which reduces the constant part of the above discriminant to the discriminant of the polynomial $\widetilde{P}(u)$ below), the expression should possibly be thought of as the “image” of the polynomial

$$\widetilde{P}(u) := u^3 + \frac{a}{(\omega-1)\sqrt[3]{\omega-3}}u + \frac{b}{\omega(\omega-3)}$$

under some discriminant-like “function”.

Does the expression above look familiar, or is it perhaps some determinant involving the coefficients in $\widetilde{P}(u)$, and $z$? Alternatively, do you see some way to attack $\textrm{Discriminant}_u(\omega P(u) +(z-u)P'(u))$ directly?


  [1]: https://mathoverflow.net/questions/379712/discriminant-of-alpha-pu-z-u-pu