  A fractional power interpretation of the Laguerre derivative $(DxD)^\alpha,\
D\equiv {d\over dx} $ is discussed. The corresponding fractional integrals are
introduced. Mapping and semigroup properties, integral representations and
Mellin transform analysis are presented. A relationship with the
Riemann-Liouville fractional integrals is demonstrated. Finally, a second kind
integral equation of the Volterra-type, involving the Laguerre fractional
integral is solved in terms of the double hypergeometric type series as the
resolvent kernel.
