A simple derivation of the addition theorems of the irregular solid harmonics, the Helmholtz harmonics, and the modified Helmholtz harmonics

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Abstract
In this article a simple derivation of the addition theorems of the irregular solid harmonics, the Helmholtz harmonics, and the modified Helmholtz harmonics is presented. Our derivation is based upon properties of the differential operator 𝒴ml(∇), which is obtained from the regular solid harmonic 𝒴ml(r) by replacing the Cartesian components of r by the Cartesian components of ∇. With the help of this differential operator 𝒴ml(∇), which is an irreducible spherical tensor of rank l, the addition theorems of the anisotropic functions are obtained by differentiating the addition theorems of the isotropic functions. The performance of the necessary differentiations is greatly facilitated by a systematic exploitation of the tensorial nature of the differential operator 𝒴ml(∇).