Geodesic
Transformation of irregular solid spherical harmonics at parallel translation of the coordinate system
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 159 (2017) no. 1, pp. 5-12

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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.
Keywords: solid spherical harmonics, parallel translation. 
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     author = {A. A. Aganin and A. I. Davletshin},
     title = {Transformation of irregular solid spherical harmonics at parallel translation of the coordinate system},
     journal = {U\v{c}\"enye zapiski Kazanskogo universiteta. Seri\^a Fiziko-matemati\v{c}eskie nauki},
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     publisher = {mathdoc},
     volume = {159},
     number = {1},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZKU_2017_159_1_a0/}
}
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A. A. Aganin; A. I. Davletshin. Transformation of irregular solid spherical harmonics at parallel translation of the coordinate system. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 159 (2017) no. 1, pp. 5-12. http://geodesic.mathdoc.fr/item/UZKU_2017_159_1_a0/