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In this paper we study fractional powers of the Bessel differential operatordefined on a semiaxis. Some important properties of such fractional powers ofthe Bessel differential operator are proved. They include connections withLegendre functions for kernel representations, fractional integral operators ofLiouville and Saigo, Mellin transform and index laws. Possible applications areindicated to differential equations with fractional powers of the Besseldifferential operator.

On fractional powers of the Bessel operator on a semiaxis