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..
Best Answer
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.