Generating Functions for Spherical Harmonics Part I: Three-Dimensional Harmonics

A new approach to the theory of spherical harmonics and Legendre functions of integer degree and order is presented, based on a generating function \[ζ ({ r}) = exp (z - iy) = ∑ {r^l P_l^m } (cos θ ){{eimφ }}{{(l + m)!}}\] and modifications thereof. The analytic properties of the harmonics, their recurrence relations and addition theorems under translation are derived in a simple and consistent way; the mathematical apparatus required is minimal. Integrals of products of surface harmonics over the unit sphere can be studied by integrating products of generating functions; with two factors this yields the orthogonality relations and normalization constants. The case of three factors is best discussed in terms of coefficients, closely related to Wigner’s $3j$-symbols, but which are integers of moderate size and satisfy a 3-term recurrence relation analogous to that of the binomial coefficients. The Regge symmetries are obtained for modified coefficients which are rational, but not integer.