Clausen’s identity for Legendre polynomials has the form (see, for example,
A generating function of the squares of Legendre polynomials, by Wadim Zudilin: <https://arxiv.org/abs/1210.2493>)
$$P\_n(\cos{\theta})^2=\sum\_{k=0}^n\frac{(-1)^k}{2^{2k}}\binom{n}{k}\binom{n+k}{n}\binom{2k}{k}\sin^{2k}{\theta}.$$
Do the analogous identity exist for the square of associated Legendre polynomials $[P\_n^l(\cos{\theta})]^2$?



Clausen’s identity for associated Legendre polynomials
There is a similar formula 
$$
\small{\left(P\_n^m(\cos\theta)\right)^2=(\sin{\theta})^{2m}\frac{(m+n)!}{(n-m)!}\sum\_{k=0}^{n-m}\frac{(-1)^k}{4^{k+m}}\binom{n+m}{k+2m}\binom{n+k+m}{n+m}\binom{2k+2m}{k+m}\sin^{2k}{\theta}}
$$
obtained from the following representation for associated Legendre polynomials
$$
P\_n^m(z)=(-1)^m \left(\frac{1-z}{1+z}\right)^{\frac{m}{2}} \frac{(m+n)!}{m! (n-m)!} \, \_2F\_1\left(-n,n+1;m+1;\frac{1-z}{2}\right)
$$
and the formula (6.1) from the paper W.N. Bailey, *[Some Theorems Concerning Products of Hypergeometric Series](http://onlinelibrary.wiley.com/doi/10.1112/plms/s2-38.1.377/full)*, Proc. London Math. Soc. **38**, 377–384 (1935):
$$
{}\_2F\_1(a,b;c;x){}\_2F\_1(a,b;a+b-c+1;x)={}\_4F\_3\left({a,b,\tfrac{a+b}2,\tfrac{1+a+b}2\atop a+b,c,a+b-c+1};4x(1-x)\right).
$$


