Three-Dimensional Addition Theorem for Arbitrary Functions Involving Expansions in Spherical Harmonics

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Abstract
For any vector r = r1 + r2 an expansion is derived for the product of a power rN of its magnitude and a surface spherical harmonic YLM(ϑ, φ) of its polar angles in terms of spherical harmonics of the angles (ϑ1, φ1) and (ϑ2, φ2). The radial factors satisfy simple differential equations; their solutions can be expressed in terms of hypergeometric functions of the variable (r</r>)2, and the leading coefficients by means of Gaunt's coefficients or 3j symbols. A number of linear transformations and three‐term recurrence relations between the radial function are derived; but in contrast to the case L = 0, no generally valid expressions symmetric in r1 and r2 could be found. By interpreting the terms operationally, an expansion is derived for the product of YLM(ϑ, φ) and an arbitrary function f(r). The radial factors are expansions in derivatives of f(r>); for spherical waves, they factorize into Bessel functions of r1 and r2 in agreement with the expansion by Friedman and Russek. The 3j symbols are briefly discussed in an unnormalized form; the new coefficients are integers, satisfying a simple recurrence relation through which they can be arranged on a five‐dimensional generalization of Pascal's triangle.

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