Mittag-Leffler

The Mittag-Leffler function $E_{\alpha, \beta}$ is defined for $z, \alpha, \beta \in \mathbb{C}$ with $\mathrm{Re}(\alpha), \mathrm{Re}(\beta) > 0$ by the series $$E_{\alpha, \beta}(z)=\displaystyle\sum_{k=0}^{\infty} \dfrac{z^k}{\Gamma(\alpha k + \beta)},$$ where $\Gamma$ denotes the gamma function.

Properties

E (0,1)(z)=1/(1-z) for abs(z) less than 1
E(1,1)(z)=exp(z)
E(2,1)(z)=cosh(sqrt(z))
E(2,1)(-z^2)=cos(z)

References

• H.J. Haubold, A.M. Mathai and R.K. Saxena: Mittag-Leffler Functions and Their Applications (2011)... (next): $(1.1)$ (has $\beta=1$ and uses the notation $E_{\alpha}$) and $(1.2)$