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

For α,β0, the generalized Mittag-Leffler function E α,β (x) is defined as

E α,β (x)= k=0 x k /Γ(αk+β)·

It was proved by H. Pollard [Bull. Am. Math. Soc. 54, 1115-1116 (1948; Zbl 0033.35902)] that for 0α1 the function E α,1 (-t) is completely monotonic for t0; W. R. Schneider [Expo. Math. 14, No. 1, 3-16 (1996; Zbl 0843.60024)] improved this by showing that E α,β (-t) is completely monotone on t0 if 0<α1 and βα. In the present note the author shows that the latter result is in fact a simple consequence of the former.

33E20Functions defined by series and integrals