## Generalized Legendre differential equation

**Vanessa V. Destura**, MSU-IIT

Starting with an eigenvalue equation,

where

Also, we set

So, substituting the given values, we find

Now, by separation of variables, the equation yields to

Let . So,

Note that

So,

Thus, the associated Legendre differential equation is

Since the solutions are symmetric in , assume . So,

Using Frobenius method (power series method), we let

Now, we take the first and second total derivative of . So,

Substituting these terms to equation yields to

Hence, the recurrence relation for the coefficients is given by