When expanding a function on a sphere $f(r,\theta, \phi)=\sum_{j=1}^{\infty}\sum_{n=0}^{\infty}\sum_{m=-n}^{n}=A_{jnm}j_{n}(\lambda_{n,j}r)Y_{n,m}(\theta,\phi)$. Since what I'm asking involves the coefficients, I'll define them as well. $A_{jnm}=\frac{2}{a^3J_{n+1}^{2}(\alpha_{n+1/2,j})}\int_0^a\int_0^{2\pi}\int_0^\pi f(r,\theta,\phi)j_n(\lambda_{n,j}r)\overline{Y}_{n,m}(\theta,\phi)r^2\sin(\theta)d\theta d\phi dr$
$\overline{Y}_{n,m}(\theta,\phi)=\sqrt{\frac{(2n+1)(n-m)!}{4\pi(n+m)!}}P^m_n(cos(\theta))e^{-im\phi}$
Why is $\overline{Y}$ necessary/convenient/used as opposed to $Y$? If you know or have vague ideas but don't want to tell me hints about things I already thought about like $\overline{Y}=(-1)^mY_{n,-m}$ are slightly better than down voting and not saying anything.
Note that the only place where complex conjugation can possibly matter is in $e^{im\theta}$, the azimuthal part of $Y_{n,m}.$ But there it's crucial: $Y_{n,m}$ should be orthogonal to any other spherical harmonics, and if one didn't use complex-conjugation this wouldn't have since one would not have $\int_0^{2\pi} e^{im\phi}e^{-im'\phi}=2\pi\delta_{m,m'}$.