### ?

## Riemann-Liouville operator in weighted Lp spaces via the Jacoby series expansion

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, which gives us the 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.