A representation theorem of the spherical wave functions

Abstract

Let $\Phi_i^{\ast}$ and $\psi_i$ (i=0,l,...,n-l) are the solutions of the equations $(\square^2-\frac{n-1}{r^2})\Phi_i^{\ast}= 04 and $\square^2\psi_i =0$ respectively. In this paper it is shown that if u and v are satisfied the equations $(\square^2-\frac{n-1}{r^2})^n u = 0$ and $\square^{2n}v=0$ respectively then u and v have the representations $u=\Phi_0^{\ast}+t\Phi_1^{\ast}+...+t^{n-1}\Phi_{n-1}^{\ast}$ and $v=\psi_0+t\psi_1+...+t^{n-1}\psi_{n-1}$ where $\square^2=\frac1{r^{n-1}}\frac{\partial}{\partial r}(r^{n-1}\frac{\partial}{\partial r})-\frac{\partial^2}{\partial t^2}$