= Solution
For one <Fourier mode>, put $\mu=\hat{\mathbf k}\cdot\mathbf e$. The collisionless equation becomes
$$
\dot\Theta+ik\mu\Theta=\dot\phi-ik\mu\psi.
$$
Use the unweighted <Legendre polynomial> expansion specified in the paper: $\Theta=\sum_{j\ge0}(-i)^j\Theta_jP_j(\mu)$. The <Legendre polynomial recurrence relation> says
$$
\mu P_j=\frac{j+1}{2j+1}P_{j+1}+\frac{j}{2j+1}P_{j-1}.
$$
The coefficient of $P_\ell$ in $ik\mu\Theta$ receives contributions from $j=\ell-1$ and $j=\ell+1$. Dividing by $(-i)^\ell$ gives respectively $-k\ell\Theta_{\ell-1}/(2\ell-1)$ and $k(\ell+1)\Theta_{\ell+1}/(2\ell+3)$. The metric sources are $\dot\phi P_0$ and $-ik\psi P_1$, the latter becoming $k\psi$ after division by $(-i)$. Hence the <neutrino Boltzmann hierarchy> is
$$
\boxed{\dot\Theta_\ell+k\left(\frac{\ell+1}{2\ell+3}\Theta_{\ell+1}-\frac{\ell}{2\ell-1}\Theta_{\ell-1}\right)
=\delta_{\ell0}\dot\phi+\delta_{\ell1}k\psi}.
$$
The lower-neighbour term is absent for $\ell=0$. The low moments are
$$
\dot\Theta_0+\frac{k}{3}\Theta_1=\dot\phi,\qquad
\dot\Theta_1+k\left(\frac25\Theta_2-\Theta_0\right)=k\psi,\qquad
\dot\Theta_2+k\left(\frac37\Theta_3-\frac23\Theta_1\right)=0.
$$
These coefficients depend on the expansion convention. In the convention with a $(2\ell+1)$ factor, the temperature multipoles are $\Theta_\ell/(2\ell+1)$; using those multipoles without converting them would give incorrect factors in the <scalar neutrino anisotropic stress>.
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