Product of $\cos\theta$ or $\sin\theta$ and a Spherical Harmonic

Question

To mix an orientation vector in spherical coordinates with a Spherical Harmonic, I am trying to find an expression for the product of $\cos\phi\sin\theta$, $\sin\phi\sin\theta$ or $\cos\theta$ with a spherical harmonic function $Y_l^m(\theta,\phi)$. What I found so far, it that is seems like this product can be expanded into no more than 2 spherical harmonics in the case of $\cos\theta Y_l^m(\theta,\phi)$ or no more than 4 spherical harmonics in the case of $\sin\phi\sin\theta Y_l^m(\theta,\phi)$ and $\cos\phi\sin\theta Y_l^m(\theta,\phi)$. It seems like:

$$\cos\theta Y_l^m(\theta,\phi)=c_+(l,m) Y_{l+1}^m(\theta,\phi)+c_-(l,m) Y_{l-1}^m(\theta,\phi)$$

(so long as $Y_{l-1}^m(\theta,\phi)$ exists, that is $l-1\geq|m|$)

and

$$\sin\phi\sin\theta Y_l^m(\theta,\phi)=c_{++}(l,m) Y_{l+1}^{m+1}(\theta,\phi)+c_{+-}(l,m) Y_{l+1}^{m-1}(\theta,\phi)+c_{-+}(l,m) Y_{l-1}^{m+1}+c_{--}(l,m) Y_{l-1}^{m-1},$$ $$\cos\phi\sin\theta Y_l^m(\theta,\phi)=c'_{++}(l,m) Y_{l+1}^{m+1}(\theta,\phi)+c'_{+-}(l,m) Y_{l+1}^{m-1}(\theta,\phi)+c'_{-+}(l,m) Y_{l-1}^{m+1}+c'_{--}(l,m) Y_{l-1}^{m-1}$$

(again, so long as all spherical harmonics exists)

I can't seem to find the analytical expressions for $c_+(l,m)$, $c_-(l,m)$, $c_{++}(l,m)$, $c_{+-}(l,m)$, $c_{-+}(l,m)$, $c_{--}(l,m)$, $c'_{++}(l,m)$, $c'_{+-}(l,m)$, $c'_{-+}(l,m)$ and $c'_{--}(l,m)$.

Does anyone know how to come up with closed expressions for these prefactors?

Answer 1

Write $\cos\theta$ as a multiple of $Y_1^0$, $\sin\theta \cos\phi$ and $\sin\theta \sin\phi$ as a linear combination of $Y_1^1$ and $Y_1^{-1}$, and then use the standard expansion of products of spherical harmonics with the Clebsch Gordan coefficients or Wigner coefficients. The selection rules of these coefficients will filter the surviving terms of $Y_{l\pm 1}$

Answer 2

Indeed, the solution strategy suggested in the answer by R.J. Mather seems to have worked out. I will accept his answer, but write the resulting expression from my derivations here as well for future reference:

\begin{align} \sin\theta\cos\phi Y_l^m&=\sum_{L=l-1}^{l+1} \sqrt{\frac{2l+1}{2(2L+1)}}C_{10l0}^{L0}\left(-C_{11lm}^{L,m+1}Y_L^{m+1}+C_{1-1lm}^{L,m-1}Y_L^{m-1}\right),\\ \sin\theta\sin\phi Y_l^m&=\sum_{L=l-1}^{l+1} i\sqrt{\frac{2l+1}{2(2L+1)}}C_{10l0}^{L0}\left(C_{11lm}^{L,m+1}Y_L^{m+1}+C_{1-1lm}^{L,m-1}Y_L^{m-1}\right),\\ \cos\theta Y_l^m&=\sum_{L=l-1}^{l+1} \sqrt{\frac{2 l+1}{2 L+1}} C_{10l0}^{L0} C_{10lm}^{Lm} Y_L^{m}, \end{align}

Note that the $L=l$ term always vanishes as a consequence of the first CG coefficient (but I could not come up with a shorter notation to eliminate this term from the sum, so just left the expression like this)