The Rodrigues’ formula for the Legendre polynomials is a mathematical expression that provides a method to calculate any Legendre polynomial using differentiation.

It is given by

with the first few Legendre polynomials being .

To derive eq348, let . The derivative of with respect to is or equivalently,

Using Leibniz’s theorem to differentiate eq349 by times with respect to gives

Comparing eq350 with eq332 reveals that is a solution to the Legendre differential equation. However, this solution does not produce the conventional Legendre polynomials of and so on.

###### Question

Prove by induction that for , and hence, .

###### Answer

For , we have . Assuming that is true, then,

So, holds for all . Similarly, . If we let , then .

To derive the factor in eq348, we again use Leibniz’s theorem to find the -th derivative of with respect to :

If we evaluate at , only the term in survives because the binomial theorem defines , resulting in

Since the conventional Legendre polynomials are defined by the condition , the solution to the Legendre differential equation in terms of must be , which is equivalent to eq348.