I have an associated Legendre Polynomial $\left(P_l^m(\cos(\theta))\right)^2$ (where $l$ and $m$ are nonnegative integers). I need to find a way to express it in terms of simple associated Legendre Polynomials in the following way
$$\left(P_l^m(\cos(\theta))\right)^2=\sum_{i=0}^\infty c_i P_i^0(\cos(\theta)),$$
and
$$\left(P_l^m(\cos(\theta))\right)^2=\sum_{i=2m}^\infty d_i P_i^{2m}(\cos(\theta)).$$
I know that the sets $\left\{P_i^0(\cos(\theta))\right\}_{i=0}^\infty$ and $\left\{P_i^{2m}(\cos(\theta))\right\}_{i=0}^\infty$ are orthogonal (with respect to a suitable inner-product) and I know that the full set $\left\{P_l^m(\cos(\theta))\right\}_{0\leq l,m< \infty}$ is complete, but what can be said about the expansions I am trying to do? I know the explicit expression of the coefficients (since I am working with the suitable inner product I talked about), but I don't know if the equalities hold.
EDIT: I know that when $m=0$, then $P_l^m(\cos(\theta))$ is a Legendre polynomial and Legendre polynomials are complete, so the first equality must hold for suitable values of $c_i$ but I still don't know if the second expansion can be done.