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We investigate fractional moments and expectations of power means ofcomplex-valued random variables by using fractional calculus. We deal with bothnegative and positive orders of the fractional derivatives. The one-dimensionaldistributions are characterized in terms of the fractional moments without anymoment assumptions. We explicitly compute the expectations of the power meansfor both the univariate Cauchy distribution and the Poincar\'e distribution onthe upper-half plane. We show that for these distributions the expectations areinvariant with respect to the sample size and the value of the power.

Power means of random variables and characterizations of distributions via fractional calculus