Here's a well-explained article about fractional derivatives and integrals (1). One freaky outcome is that the factorials of fractional numbers can be found. (Example: negative one half factorial = square root of pi.) It can be applied to all things quantum mechanical (2) so there's a fractional Schrödinger equation etc. I had to look it up, but the links are worth the bother.

(1) https://www.mathpages.com/home/kmath616/kmath616.htm (2) https://arxiv.org/abs/1009.5533

https://www.mathpages.com/home/kmath616/kmath616.htm