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Some analytical and numerical properties of the Mittag-Leffler functions. (English) Zbl 1312.33057

In the paper two results are obtained. First, the authors partly prove Mainardi’s conjecture:
for every \(t > 0\) and for every fixed \(\alpha\), \(0 < \alpha < 1\), the following estimate holds: \[ \frac{1}{1 + t^\alpha \Gamma(1 - \alpha)} \leq e_{\alpha}(t) \leq \frac{1}{1 + \frac{t^\alpha}{\Gamma(1 + \alpha)}}. \eqno{(1)} \]
The authors prove that the inequality (1) holds at least in a neighbourhood of \(t = 0\).
Second, an approximate numerical algorithm is proposed for the evaluation of values of the Mittag-Leffler function of a real negative argument \[ e_{\alpha}(t) := E_{\alpha}(- t^{\alpha}) = \sum\limits_{n=0}^{\infty} (-1)^n \frac{t^{\alpha n}}{\Gamma(\alpha n + 1)}. \]
The algorithm is based on Diethelm’s predictor-corrector algorithm (as described, e.g., in [K. Diethelm et al., Numer. Algorithms 36, No. 1, 31–52 (2004; Zbl 1055.65098)]) The convergence is proved. The results are illustrated by corresponding plots, comparison with other numerical procedures are given.

MSC:

33E12 Mittag-Leffler functions and generalizations
34A08 Fractional ordinary differential equations

Citations:

Zbl 1055.65098

Software:

DLMF
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References:

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