In mathematics, the Mittag-Leffler function ${\displaystyle E_{\alpha ,\beta }}$ is a special function, a complex function which depends on two complex parameters ${\displaystyle \alpha }$ and ${\displaystyle \beta }$. It may be defined by the following series when the real part of ${\displaystyle \alpha }$ is strictly positive:

${\displaystyle E_{\alpha ,\beta }(z)=\sum _{k=0}^{\infty }{\frac {z^{k}}{\Gamma (\alpha k+\beta )}},}$

where ${\displaystyle \Gamma (x)}$ is the gamma function. When ${\displaystyle \beta =1}$, it is abbreviated as ${\displaystyle E_{\alpha }(z)=E_{\alpha ,1}(z)}$. For ${\displaystyle \alpha =0}$, the series above equals the Taylor expansion of the geometric series and consequently ${\displaystyle E_{0,\beta }(z)={\frac {1}{\Gamma (\beta )}}{\frac {1}{1-z}}}$.

In the case ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are real and positive, the series converges for all values of the argument ${\displaystyle z}$, so the Mittag-Leffler function is an entire function. This function is named after Gösta Mittag-Leffler. This class of functions are important in the theory of the fractional calculus.

For ${\displaystyle \alpha >0}$, the Mittag-Leffler function ${\displaystyle E_{\alpha ,1}(z)}$ is an entire function of order ${\displaystyle 1/\alpha }$, and is in some sense the simplest entire function of its order.

The Mittag-Leffler function satisfies the recurrence property (Theorem 5.1 of )

${\displaystyle E_{\alpha ,\beta }(z)={\frac {1}{z}}E_{\alpha ,\beta -\alpha }(z)-{\frac {1}{z\Gamma (\beta -\alpha )}},}$

from which the Poincaré asymptotic expansion

${\displaystyle E_{\alpha ,\beta }(z)\sim -\sum _{k=1}{\frac {1}{z^{k}\Gamma (\beta -k\alpha )}}}$

follows, which is true for ${\displaystyle z\to -\infty }$.