Context: ''Handbook of Mathematical Functions'' by Milton Abramowitz and Irene A. Stegun.
I've come across this in chapter 3: Elementary Analytical Methods:
"If $z^n = u_n + i v_n$, then $z^{n + 1} = u_{n + 1} + i v _{n + 1}$ where $$3.7.23 \qquad u_{n + 1} = x u_n - y v_n, v_{n + 1} = x v_n + y u_n$$ $\mathscr R \, z_n$ and $\mathscr I \, z_n$ are called harmonic polynomials."
Earlier $z$ has been defined as $x + i y$, and we have also been given:
$$3.7.22 \qquad z_n = [x^n - \dbinom n 2 x^{n - 2} y^2 + \dbinom n 4 x^{n - 4} y^4 - \cdots] + i [\dbinom n 1 x^{n - 1} y - \dbinom n 3 x^{n - 3} y^3 + \cdots]$$
Is this supposed to be a definition of either a "harmonic polynomial", or are they "instances" of "harmonic polynomials", or is it somehow a way of defining the relationship between two sets of polynomials, as in "a pair of polynomials which are harmonic with respect to each other" or something?
Wikipedia gives the definition of a harmonic polynomial "multivariate polynomials whose laplacian is zero" which I sort of get, but I'm not sure if that's in fact the same definition as is being used in Abramowitz and Stegun.