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A note on the complete monotonicity of the generalized Mittag-Leffler function. (English) Zbl 0964.33011

For $\alpha ,\beta \ge 0$, the generalized Mittag-Leffler function ${E}_{\alpha ,\beta }\left(x\right)$ is defined as

${E}_{\alpha ,\beta }\left(x\right)=\sum _{k=0}^{\infty }{x}^{k}/{\Gamma }\left(\alpha k+\beta \right)·$

It was proved by H. Pollard [Bull. Am. Math. Soc. 54, 1115-1116 (1948; Zbl 0033.35902)] that for $0\le \alpha \le 1$ the function ${E}_{\alpha ,1}\left(-t\right)$ is completely monotonic for $t\ge 0$; W. R. Schneider [Expo. Math. 14, No. 1, 3-16 (1996; Zbl 0843.60024)] improved this by showing that ${E}_{\alpha ,\beta }\left(-t\right)$ is completely monotone on $t\ge 0$ if $0<\alpha \le 1$ and $\beta \ge \alpha$. In the present note the author shows that the latter result is in fact a simple consequence of the former.

##### MSC:
 3.3e+21 Functions defined by series and integrals
##### Keywords:
Mittag-Leffler function; complete monotonicity