Transformation of Irregular Solid Spherical Harmonics with Parallel Translation of the Coordinate System

Abstract

Solid spherical harmonics and spherical functions are widely used for studying physical phenomena in spatial domains bounded by spherical or nearly-spherical surfaces. In this case, it is frequently needed to transform these functions with a parallel translation of the coordinate system. Specifically, this scenario arises in describing the hydrodynamic interaction of spherical or weakly-nonspherical gas bubbles in the unbounded volume of an incompressible fluid. In the two-dimensional (axisymmetric) case, when Legendre polynomials act as spherical functions, the transformation can be conducted with a well-known compact expression. In the three-dimensional case, similar well-known expressions are rather complex (for example, the Clebsch–Gordan coefficients are used in these expressions), which makes their use difficult. This paper describes a derivation of such an expression that naturally leads to a compact form of the respective coefficients. Actually, these coefficients are a generalization to the three-dimensional case of similar well-known coefficients in the two-dimensional (axisymmetric) case.