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

For \(\alpha ,\beta \geq 0\), the generalized Mittag-Leffler function \(E_{\alpha ,\beta }(x)\) is defined as \[ E_{\alpha ,\beta }(x)=\sum _{k=0} ^\infty x^k/\Gamma (\alpha k+\beta). \] It was proved by H. Pollard [Bull. Am. Math. Soc. 54, 1115-1116 (1948; Zbl 0033.35902)] that for \(0\leq \alpha \leq 1\) the function \(E_{\alpha ,1}(-t)\) is completely monotonic for \(t\geq 0\); W. R. Schneider [Expo. Math. 14, No. 1, 3-16 (1996; Zbl 0843.60024)] improved this by showing that \(E_{\alpha ,\beta }(-t)\) is completely monotone on \(t\geq 0\) if \(0<\alpha \leq 1\) and \(\beta \geq \alpha \). In the present note the author shows that the latter result is in fact a simple consequence of the former.


33E20 Other functions defined by series and integrals