Weird Expansion: Fractional Power Series

[zetafunction]

Dos this fractional Taylor series

$$(a+x)^{-r} = \sum_{m=-\infty}^{\infty} \frac{ \Gamma (-r+1)}{\Gamma (m+\alpha+1) \Gamma(-r-n-\alpha+1)}a^{(-r-m-\alpha )}x^{m+\alpha}$$

makes sense for x < 1 or x >1 and alpha being an arbitrary real number ..for example ?? , here a and r are real numbers , the idea here is if we can define a fractional power series expansion generalizing the usuarl Laurent or Taylor series.

 

[g_edgar]

I tried $r= 1,\alpha=1/2,a=1,x=1/2$.
The series diverges since the term does not go to zero as $m \to -\infty$

P.S. You have misprint $n$ for $m$ , right?