Recursive computation of spherical harmonic rotation coefficients of large degree

Understand

Computation of the spherical harmonic rotation coefficients or elements of Wigner's d-matrix is important in a number of quantum mechanics and mathematical physics applications.

• Particularly, this is important for the Fast Multipole Methods in three dimensions for the Helmholtz, Laplace and related equations, if rotation-based decomposition of translation operators are used.
• In these and related problems related to representation of functions on a sphere via spherical harmonic expansions computation of the rotation coefficients of large degree $n$ (of the order of thousands and more) may be necessary.
• Existing algorithms for their computation, based on recursions, are usually unstable, and do not extend to $n$.