primitive recursion

Noun
$$\begin{align} \rho(g, h) (0,x_1,\ldots,x_k) &= g(x_1,\ldots,x_k) \\ \rho(g, h) (y+1,x_1,\ldots,x_k) &= h(y,\rho(g, h) (y,x_1,\ldots,x_k),x_1,\ldots,x_k)\,\end{align}$$.
 * 1)  Recursion to a fixed depth.
 * 2)  The operator $$\rho\,$$ that creates a new function from two functions $$g\,$$ and $$h\,$$ such that: