Here is a normal quadratic parabola:
$f_{even}=\frac{x^2}{4}+\frac{\pi^2}6$
Here is an odd parabola:
$f_{odd}= \text{Li}_2\left(e^x\right)+ x \log \left(1-e^x\right)-\left(\frac{x^2}{4}+\frac{\pi ^2}{6}\right)$
The difference between them goes to zero as $x\to\infty$
Here is a normal cubic parabola (odd):
$f_{odd}=\frac{x^3}{12}+\frac{\pi ^2 x}{6}$
Here is an even cubic parabola:
$f_{even}=2 \text{Li}_3\left(e^x\right)-x \text{Li}_2\left(e^x\right)-\left(\frac{x^3}{12}+\frac{\pi ^2 x}{6}\right)$
They approach each other at infinity:
So, what properties can have those counterparts of normal polynomials but with the opposite parity? Are there properties, common with polynomials or conic sections?