The polynomials $P_l^m$ are the associated Legendre polynomials,
whereas $P_l$ are Legendre polynomials. The formula you gave are their definitions. $P_l^m$ is defined as in the formula, these polynomials are not the m-th derivatives (or powers or whatever) of $P_l$.
The polynomials $P_l^m$ are the associated Legendre polynomials, whereas $P_l$ are Legendre polynomials. The formula you gave are their definitions. $P_l^m$ is defined as in the formula, these polynomials are not the m-th derivatives (or powers or whatever) of $P_l$.