greetings . the mittag-leffler function is defined as :
$E_{\alpha}(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{\Gamma(1+\alpha n)}$ , $\Re(\alpha)>0$ , $z\in \mathbb{C} $
it can be continued for $\Re(\alpha)\leq0 $ via :
$E_{\alpha}(z)=-\sum_{n=1}^{\infty}\frac{z^{-n}}{\Gamma(1-\alpha n)}$
or
$E_{-\alpha}(z)=1-E_{\alpha}(z^{-1})$
according to
http://www.hindawi.com/journals/jam/2011/298628/ (9.2)et al .
however , there seems to be a discrepancy in the literature . for instance, according to
http://arxiv.org/pdf/math-ph/0406047 (8)
the following relation appears :
$E_{\alpha}(z)=\sum_{n=1}^{\infty}\frac{(-z)^{-n}}{\Gamma(1-\alpha n)}$, $|z|>1$
which one of the two relations is the correct one !?!?
thanks in advance .
