In evaluating a discriminant related to a dynamical system based on the polynomial $P(z) = z^3 + az + b$, I recently came across the following expression 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, $\omega$ is a real parameter (with $\omega\in(0,1)\cup(1,3)$) associated with the order of a fractional derivative, and $\lambda:= \sqrt[3]{\omega-3}.$

The above expression may possibly best be thought of as a function (perhaps something like a determinant involving $z$ and the coefficients of) the polynomial $$\widetilde{P}(z) := z^3 + \frac{a}{(\omega-1)\sqrt[3]{\omega-3}}z + \frac{b}{\omega(\omega-3)}.$$

The above expression can also be transformed into

$$3az^3 + 3b\omega z^2 - a^2(\omega(\omega-2)-2)z - 3ab\omega(\omega-3)$$

with the change of variables $a\mapsto a(\omega-1)\sqrt[3]{\omega-3},\,\,b\mapsto b\omega(\omega-3)$ and a change of sign.

Do the above expressions or any of their terms look familiar?