Harmonic Polynomials

Abstract

Let us consider a d-dimensional Euclidian space with Cartesian coordinates x₁,x₂,……,x_d.In this space, we can define a hyperradius r by the relationship:

$$ {{r}^{2}} = \sum\limits_{{j = 1}}^{d} {x_{j}^{2}} $$

(1-1)

We can also define the generalized Laplacian operator Δ by

$$ \Delta = \sum\limits_{{j = 1}}^{d} {\frac{{{{\partial }^{2}}}}{{\partial x_{j}^{2}}}} $$

(1-2)

A homogeneous polynomial of order n in the coordinates x₁,x₂,……,x_d. is defined to be a polynomial of the form:

$$ {{f}_{n}} = Ax_{1}^{{{{n}_{1}}}}x_{2}^{{{{n}_{2}}}} \ldots x_{d}^{{{{n}_{d}}}} + Bx_{1}^{{{{{n'}}_{1}}}}x_{2}^{{{{{n'}}_{2}}}} \ldots x_{d}^{{{{{n'}}_{d}}}} + \ldots $$

(1-3)

where A, B, C, etc are constants, and

$$ \begin{gathered} {{n}_{1}} + {{n}_{2}} + \ldots + {{n}_{d}} = n \hfill \\ {{{n'}}_{1}} + {{{n'}}_{2}} + \ldots + {{{n'}}_{d}} = n{\text{ etc}}{\text{.}} \hfill \\ \end{gathered} $$