How to program Legendre Polynomial for Generalized Spherical Harmonics?

Hi All,

I am looking for some help on how to program/calculate the Legendre polynomial for generalized spherical harmonics.. I have been using the built-in functions legendre and legendreP with great success for some applications, but now with generalized spherical harmonics (Seen below) I need to add an extra term.

I don't know how matlab calculates legendre.m but I want to avoid using syms if possible. As I've been told it might make the calculations incredibly slow.

Any advice or help on how to go about this?

Thanks!

edit: title