On the fractional Legendre equation and fractional Legendre functions

In this paper we propose a fractional generalization of the well-known Legendre equation. We obtain a solution in the form of absolutely convergent power series with radius of convergence 1. We then truncate the power series to obtain the even and odd fractional Legendre functions in closed forms. These functions converge to the Legendre polynomials as the fractional derivative approaches 1, and new explicit formulas of the even and odd Legendre polynomials have been derived.