Comparison with Mathematica

In the "Applications" section of Mathematica's documentation page for WignerD, the rotation matrix is constructed from Euler angles \((\psi,\theta,\phi)\) according to the expression

RotationMatrix[-phi, {0, 0, 1}].RotationMatrix[-theta, {0, 1, 0}].RotationMatrix[-psi, {0, 0, 1}]

This is the inverse of the matrix given by

RotationMatrix[psi, {0, 0, 1}].RotationMatrix[theta, {0, 1, 0}].RotationMatrix[phi, {0, 0, 1}]

The latter, of course, would be equivalent to a rotor \(e^{\psi\vec{z}/2}\, e^{\theta\vec{y}/2}\, e^{\phi\vec{z}/2}\), which is what I would have denoted \(\mathbf{R}_{(\psi, \theta, \phi)}\). So my rotor gives the inverse rotation of Mathematica's Euler angles. This could also be viewed as a disagreement over active and passive transformations.

However, there are still other disagreements. The same page states that

WignerD[{j,m1,m2},psi,theta,phi]

gives the function \(\mathfrak{D}^j_{m_1,m_2}(\psi,\theta,\phi)\), and the spherical harmonics are related to \(\mathfrak{D}\) by