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A note on property of the Mittag-Leffler function. (English) Zbl 1194.30002

Summary: Recently the authors have found in some publications that the following property \((0,1)\) of the Mittag-Leffler function is taken for granted and used to derive other properties
\[ E_\alpha\big(a(t+s)^\alpha\big)=D_\alpha\big(at^\alpha\big)E_\alpha\big(as^\alpha\big),\qquad t,s\geq 0,\tag{0.1} \]
where \(a\) is a real constant and \(\alpha>0\). In this note, we prove that the above property is unavailable unless \(\alpha=1\) or \(a=0\). Moreover, a new equality for \(E_\alpha(at^\alpha)\) is developed, whose limit state as \(\alpha\uparrow 1\) is just the propertiy \((0,1)\).

MSC:

30B99 Series expansions of functions of one complex variable
26A33 Fractional derivatives and integrals
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References:

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