  In this paper we use the orthogonal system of the Jacobi polynomials as a
tool to study the Riemann-Liouville fractional integral and derivative
operators on a compact of the real axis.This approach has some advantages and
allows us to complete the previously known results of the fractional calculus
theory by means of reformulating them in a new quality.
  The proved theorem on the fractional integral operator action is formulated
in terms of the Jacobi series coefficients and is of particular interest. We
obtain a sufficient condition for a representation of a function by the
fractional integral in terms of the Jacobi series coefficients. We consider
several modifications of the Jacobi polynomials what gives us an opportunity to
study the invariant property of the Riemann-Liouville operator. In this
direction we have shown that the fractional integral operator, acting in the
weighted spaces of Lebesgue square integrable functions, has a sequence of the
included invariant subspaces.
