Below I will give some definitions. My question is: do these appear in the literature, and if so, under what name?

Let $G$ and $H$ be groups that may not be commutative. For $y\in G$, define $R_y:G\to G$ by $R_y(x)=xy$. Let $f$ be a function from $G$ to $H$. Define $\delta_n(f):G^n\to H$ by \begin{align*} \delta_0(f) &= 1 \\\\ \delta_1(f)(x) &= f(1) f(x)^{-1} \\\\ \delta_2(f)(x,y) &= f(1) f(x)^{-1} f(xy) f(y)^{-1} \\\\ \delta_3(f)(x,y,z) &= f(1) f(x)^{-1} f(xy) f(y)^{-1} f(yz) f(xyz)^{-1} f(xz) f(z)^{-1} \end{align*} and in general $$ \delta_{n+1}(f)(x_1,\dotsc,x_{n+1}) = \delta_n(f)(x_1,\dotsc,x_n) \delta_n(f\circ R_{x_{n+1}})(x_1,\dotsc,x_n)^{-1}. $$ This has one term for each subset $J\subseteq\{1,\dotsc,n\}$, with exponent $(-1)^{|J|}$. The order of the terms corresponds to the Binary Reflected Gray Code (see Wikipedia, for example). One could imagine using other orders such as lexicographic, but the BRGC order seems to do the right thing for the examples that I am considering.

I'll say that $f$ is *polynomial of degree at most $n$* if $\delta_{n+1}(f)$ is the constant function with value $1$. Clearly $f$ is polynomial of degree at most $0$ iff it is constant, and it is polynomial of degree at most $1$ iff it is a constant times a homomorphism. The commutative case is fairly well-known, and is consistent with the usual meaning of 'polynomial' for maps $\mathbb{Z}^p\to\mathbb{Z}^q$. I know of a 1971 paper by Andreas Dress, but it would not surprise me if there were earlier references. However, I have never seen the noncommutative case.

Even in the commutative case, it takes some work to prove that any composite of polynomial maps is polynomial. I do not know whether that holds in the noncommutative case.