The paper is devoted to study the generalized fractional calculus of arbitrary complex order for the H -function defined by Inayat Hussain . The classical fractional integrals and derivatives of Riemann-Liouville type are treated. The considered generalized fractional integration and differentiation operators contain the Gauss Hypergeometric function as a kernel and generalize classical Riemann- Liouville, Erdelyi-Kober types ones. It is proved that the generalized fractional integrals and derivatives of H -function turn also out H -functions but of greater order. Especially, the obtained results define more precise and general ones than known. Corresponding assertion for Riemann-Liouville and Erdelyi-Kober fractional integral operators are also presented.