Only when both indices are zero can the Wigner D-matrix be written in terms of these functions.
Only when both indices are zero can the Wigner D-matrix be written in terms of these functions. For scattering from a central potential, a partial wave expansion in terms of these functions can be done for the wave function and scattering amplitude. Expanding the reciprocal of the root of quantity one, minus two a x, plus a squared, as a power series yields these functions multiplied by a power of a; that reciprocal is a generating function that appears in some (*) multipole expansions. They are scaled by the square root of two l plus one over four pi in a definition of the spherical harmonics for m equals zero. The nth of them is equal to the reciprocal of two to the n, times n factorial, all times the nth derivative of quantity x squared minus one, to the n, according to a Rodrigues formula. For 10 points, name these orthogonal polynomials that appear in solutions to Laplace's equation in spherical coordinates.
Only when both indices are zero can the Wigner D-matrix be written in terms of these functions.
Legendre polynomials [or: associated Legendre polynomials; in place of "polynomials", accept: "functions"; prompt on: "Jacobi polynomials", "spherical harmonics"]