Harmonic polynomial

A polynomial with $x_1,\ldots,x_n$ as variables that satisfies the Laplace equation. Any harmonic polynomial may be represented as the sum of homogeneous harmonic polynomials. If $n=2$, there are only two linearly independent homogeneous harmonic polynomials of degree $m$ — for example, the real and the imaginary part of the expression $(x_1+ix_2)^m$. If $n=3$, the number of linearly independent homogeneous polynomials of degree $m$ is $2m+1$. In the general case — $n\geq2$ — the number of linearly independent homogeneous harmonic polynomials of degree $m$ is

$$K_n^m-K_n^{m-2},\quad m\geq2,$$

where

$$K_n^m=\frac{n(n+1)\ldots(n+m-1)}{m!}$$

is the number of permutations of $n$ objects taken $m$ at a time with $m$ repetitions. The homogeneous harmonic polynomials, $V_m(x)$, are also known as spherical functions (in particular if $n=3$). If $n=3$, one may write, in spherical coordinates

$$V_m(x)=r^mY_m(\theta,\phi),$$

where $r=\sqrt{x_1^2+x_2^2+x_3^2}$ and $Y_m(\theta,\phi)$ is a spherical function of degree $m$.

References

[1]	S.L. Sobolev, "Partial differential equations of mathematical physics" , Pergamon (1964) (Translated from Russian) MR0178220 Zbl 0123.06508
[2]	A.N. [A.N. Tikhonov] Tichonoff, A.A. Samarskii, "Differentialgleichungen der mathematischen Physik" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian) MR104888
[3]	M. Brélot, "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris (1959) MR0106366 Zbl 0084.30903

E.D. Solomentsev

A finite linear combination of harmonics. Real-valued harmonic polynomials can be represented in the form

$$\sum_{k=1}^NA_k\sin(\omega_kx+\phi_k)$$

for a given natural number $N$, non-negative $A_k$, and real $\omega_k$, $\phi_k$, $k=1,\ldots,N$. Complex-valued harmonic polynomials can be represented in the form

$$\sum_{k=-m}^nc_ke^{i\omega_kx}$$

where $n$ and $m$ are natural numbers, $\omega_k$ is real and the $c_k$, $k=-m,-m+1,\ldots,n$, are complex. Harmonic polynomials are the simplest almost-periodic functions (cf. Almost-periodic function).