Identity for the product of two different associated Legendre polynomials

In the answer to Clausen’s identity for associated Legendre polynomials the following result was indicated: $$\small{\left(P_n^m(\cos\theta)\right)^2=(\sin{\theta})^{2m}\frac{(m+n)!}{(n-m)!}\sum_{k=0}^{n-m}\frac{(-1)^k}{4^{k+m}}\binom{n+m}{k+2m}\binom{n+k+m}{n+m}\binom{2k+2m}{k+m}\sin^{2k}{\theta}}.$$ Can it be further generalized to the product of two different associated Legendre polynomials $P_{n_1}^{m_1}(\cos\theta)P_{n_2}^{m_2}(\cos\theta)$? Such a generalization, if found, can be used in atomic transition form factor calculations.