Transformation of irregular solid spherical harmonics at parallel translation of the coordinate system

When studying physical phenomena in spatial regions bounded by spherical or slightly non-spherical surfaces, spherical functions and solid spherical harmonics are widely used. A problem of transformation of those functions and harmonics with translation of the coordinate system frequently arises. Such a situation occurs, in particular, when the hydrodynamic interaction of spherical or slightly non-spherical gas bubbles in an unbounded volume of incompressible fluid is described. In the two-dimensional (axisymmetric) case, when the role of spherical functions is played by the Legendre polynomials, such a transformation can be performed using a well-known compact expression. Similar known expressions in the three-dimensional case are rather complex (they, for example, include the Clebsch–Gordan coefficients), which makes their application more complicated. The present paper contains derivation of such an expression, naturally leading to a compact form of its coefficients. Those coefficients are, in fact, a generalization to the three-dimensional case of the analogous known coefficients in the two-dimensional (axisymmetric) case.