Given a classical Taylor expansion as $f(x)=\sum_{i=0}^{\infty} \frac{d^{i}f(x_{0})}{dx^{i}}\frac{(x-x_{0})^{i}}{i!}$ where i is a nonnegative integer.
Can we generalize this expansion to fractional number $\alpha$ which satisfied $f(x)=\sum_{i=0}^{\infty} \frac{d^{(i\alpha)}f(x_{0})}{dx^{(i\alpha)}}\frac{(x-x_{0})^{i\alpha}}{\Gamma(i\alpha+1)}$ ? I am not sure this simple generalization is correct or not?
If it is not correct, what is the coefficient $b_{i}$ which satisfy $f(x)=\sum_{i=0}^{\infty} b_{i}(x-x_{0})^{i\alpha}$ ?