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Numerical algorithm for calculating the generalized Mittag-Leffler function. (English) Zbl 1190.65033
The generalized Mittag–Leffler function $$E_{\alpha,\beta}(z)$$ is an entire function with two real parameter $$\alpha$$ and $$\beta$$ of the form
$E_{\alpha,\beta}(z)=\sum_{k=0}^\infty \frac{z^k}{\Gamma(\alpha k+\beta)},$
for $$z \in \mathbb{C}$$. A numerical algorithm for calculating of this function for arbitrary complex argument $$z$$ and real $$\alpha>0$$ and $$\beta \in \mathbb{R}$$ is presented. The algorithm uses the Taylor series, the exponentially improved asymptotic series and integral representations to obtain optimal stability and accuracy of the algorithm. Special care is applied to the limits of validity of the different schemes to avoid instabilies in the algorithm. Numerical tests and speed analysis of the algorithm is given.

##### MSC:
 65D20 Computation of special functions and constants, construction of tables 33E12 Mittag-Leffler functions and generalizations 33F05 Numerical approximation and evaluation of special functions