Two years ago in graduate school I came across an article that gave a gentle introduction to the fractional derivative $D^\alpha$ and derived some properties of the semigroup $(D^\alpha)_{\alpha\geq 0}$, including the hypercyclicity of the semigroup.
I don't remember much more than that, especially the lingo that the article used. I wish I would have saved it, but after many hours surfing the web, I thought I'd reach out to see if anyone has references to fractional derivatives, hypercyclicity, and things of that ilk.