Help with Legendre Differential equation

[zamoura]

I have never seen the Legendre Function and the Legendre function of the second kind multiplied together for a solution. Can someone point me in the right direction to learn more about solving these equations with solutions like this? Thanks very much

 

[HallsofIvy]

What you have written is pretty much unintelligble. You say "the solution is given by" but your formula includes things that are not in the equation and are not defined separtely. In particular you say "$\overline{r}_</\overline{r}_> for r,r_q$ yet there is no r or r_q in your equation.

In any case, the solutions to the Legendre's equation are, as you might suspect from the name, the Legendre functions, not products of Legendre functions.