There is a mathematical concept of polynomials with non integer degree? for example, something mid way between a linear function and a parabola.
I have interest in a general expression which can continuously vary between polynomials of integer degrees, and of course many interpolation may be defined, but there is anyone which preserves interesting properties of polynomials? Or some "canonical" or generally accepted or generally useful way to interpolate between polynomials?
Maybe there is a canonical way to generalize the product operator? $$\prod_{n=0}^{r\in \mathbb{R}}\left ( x-a_n \right )$$
Possibly a "reasonable" polynomial with the same root $a_n=a_0$ would be $(x-a_0)^r$