Let $P^k=$homogeneous polynomials of degree $k$ in $x$, $y$, $z$, $k=0, 1, 2, \dots, $, i.e. $P^k= \text{Span} \{x^{k_x}y^{k_y}z^{k_z} : k_x+k_y+k_z=k\}$. This is maybe a silly question, but I am not able to figure out what is the spherical harmonics of that kind of set? In other words, what would be another way to express $H^k= \{f \in P^k : \Delta f = 0\}$, where $\Delta$ is the usual Laplacian operator?
Thanks!