The associated Legendre polynomials are a sequence of orthogonal polynomials that are solutions to the associated Legendre differential equation:

which can also be expressed as

Eq360 can be derived from the Legendre differential equation. It begins with differentiating eq332 times using Leibniz’s theorem to give

where are the Legendre polynomials.

Assuming that , only the first three terms and the first two terms of the first sum and the second sum, respectively, survive. So, eq361 becomes

where .

Let

Substituting , and in eq362 yields eq360, where

Substituting eq348 in eq363a gives the un-normalised associated Legendre polynomials:

with the normalisation constant given by eq391.

Finally, when , eq360 becomes the Legendre differential equation. Therefore, the associated Legendre differential equation is a generalisation of the Legendre differential equation.

###### Question

If we have assumed that when we differentiate the Legendre differential equation times to obtain the associated Legendre differential equation, does it mean that in the associated Legendre differential equation are restricted to values greater than or equal to two?

###### Answer

No, the procedure merely demonstrates that the solutions to the associated Legendre differential equation are related to those of the Legendre differential equation by . The allowed values of must be consistent with the solutions to the associated Legendre differential equation. To avoid the trivial solution of , we require that . Thus, can take values of , and while we may choose to differentiate with , the full set of associated Legendre functions exists for .