\[C^{(\lambda)}_{n}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{%
2}\cos\phi\right)=\sum_{\ell=0}^{n}2^{2\ell}(n-\ell)!\frac{2\lambda+2\ell-1}{2%
\lambda-1}\frac{({\left(\lambda\right)_{\ell}})^{2}}{{\left(2\lambda\right)_{n%
+\ell}}}(\sin\theta_{1})^{\ell}C^{(\lambda+\ell)}_{n-\ell}\left(\cos\theta_{1}%
\right)(\sin\theta_{2})^{\ell}C^{(\lambda+\ell)}_{n-\ell}\left(\cos\theta_{2}%
\right)C^{(\lambda-\frac{1}{2})}_{\ell}\left(\cos\phi\right),\]