 The generating function approach is directly connected to the multipole expansion in electrostatics, as explained below, and is how the polynomials were first defined by Legendre in 1782.A third definition is in terms of solutions to Legendre's differential equation

This differential equation has regular singular points at so if a solution is sought using the standard Frobenius or power series method, a series about the origin will only converge for in general.When is an integer, the solution that is regular at is also regular at , and the series for this solution terminates (i.e.