Taylor Series in Fractional Calculus

Question

I recently studied fractional calculus, namely the possibility to define fractional derivatives of some functions, like

$$\frac{\text{d}^{1/2}}{\text{d}x^{1/2}}\ f(x) ~~~~~~~~~~~~~ \frac{\text{d}^{2/3}}{\text{d}x^{2/3}}\ f(x)$$

and so on.

Now the question that came up into my mind is: if such a construction is possible, can we built " new " Taylor series for well known function in order to take into account (some) fractional derivatives too?

I know the first problems that would arise would be: how could we take the whole possible derivatives of order between $0$ and $1$? They are infinite. And Yeah, that could be a really huge problem..

Are there any example of Fractional Taylor Series?

P.s. I've already read other similar questions, but they are too arid and I didn't find any good answer yet..

Answer 1

Yes:

Series expansion in fractional calculus and fractional differential equations

Fractional calculus and the Taylor-Riemann series

Examples: see p.7-9 of the first link and p.18 of second link.

Answer 2

Doing a few calculations, I find the following must be:

$$f^{(n)}(x)=\sum_{k=-\infty}^\infty\frac{f^{(k)}(a)x^{k-n}}{\Gamma(k-n+1)}$$

where we define ${1\over\Gamma(-n)}=0$ for $n\in\mathbb N$

The regular Taylor series doesn't hold by induction, but this one does.

You could probably adjust $k$ to fit what you want.

(note, it fails to converge for most $f(x)$)