I want to know under what conditions does the Mittag-Leffler function ${E\_{\alpha ,1}}(z),(0 < \alpha < 1)$ has no real zero, where


${E\_{\alpha ,1}}(z) = \sum\limits\_{k = 0}^\infty {\frac{{{z^k}}}{{\Gamma (\alpha k + 1)}}}$.


${E\_{1,1}}(z) = \sum\limits\_{k = 0}^\infty {\frac{{{z^k}}}{{\Gamma (k + 1)}}} {\text{ = }}{{\text{e}}^z}$ or ${E\_{0,1}}(z) = \frac{1}{{1 - z}}$ has no zero, how about $\alpha \in (0,1)$?



Under what conditions does the Mittag-Leffler function ${E_{\alpha ,1}}(z),(0 < \alpha < 1)$ has no real zero?
None of these functions have real zeroes, because they can be written as moment generating
functions of certain random variables $Y\_\alpha$. More precisely, for $0<\alpha <1$
$$E\_{\alpha,1}(z)=\mathbb{E} e^{zY\_\alpha}$$
where $$X\_\alpha:=(1/Y\_{\alpha})^{1/\alpha}$$ has the extreme positive stable distribution
with index $\alpha$ (Levy-distribution), i.e- Laplace transform 
$$Ee^{-pX\_\alpha}=e^{-p^\alpha}$$ for $p\geq 0$. See W. Feller, Intro. Prob. Theo. Appl. II,
(1971), p.453.


