Lobachevskii Journal of Mathematics

, Volume 39, Issue 3, pp 433–438 | Cite as

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

  • A. A. Aganin
  • A. I. Davletshin


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.


Solid spherical harmonics parallel translation 


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© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Institute of Mechanics and Engineering, Kazan Scientific CenterRussian Academy of SciencesKazanRussia

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