Radial multipliers in solutions of the Helmholtz equations

We define radial multipliers using solutions of the Helmholtz equation, which depend on the radial coordinate, and we find the recurrence relations between them in the space of any dimension , in which the Helmholtz operator is defined. It is shown that the procedure of differentiation of these multipliers leads to a system of solutions of the Helmholtz equation, represented as products of the radial multipliers and harmonic polynomials. Theorems about the properties of radial multipliers and the structure of harmonic polynomials in the solutions of Helmholtz equation are given. These solutions constructed using radial multipliers and harmonic polynomials are proposed to be used in gradient elasticity for multi-layered domains with spherical and cylindrical boundaries, since they allow to present boundary conditions in explicit algebraic form.