Novel Results on Legendre Polynomials in the Sense of a Generalized Fractional Derivative

Abstract

In this article, new results are investigated in the context of the recently introduced Abu-Shady–Kaabar fractional derivative. First, we solve the generalized Legendre fractional differential equation. As in the classical case, the generalized Legendre polynomials constitute notable solutions to the aforementioned fractional differential equation. In the sense of the fractional derivative of Abu-Shady–Kaabar, we establish important properties of the generalized Legendre polynomials such as Rodrigues formula and recurrence relations. Special attention is also devoted to another very important property of Legendre polynomials and their orthogonal character. Finally, the representation of a function f∈Lα2([−1,1]) in a series of generalized Legendre polynomials is addressed.