The class of primitive recursive functions

Question

Usually the scheme of primitive recursion is defined as follows: $$ h(x, 0)=f(x) \\ h(x, y+1)=g(x,y,h(x,y)) $$ I was wondering whether the class of primitive recursive functions would be smaller if we changed the second formula to $$ h(x,y+1)=g(x,h(x,y)). $$

Answer 1

Consider a function $h(x,y)$ s.t. $h(x,y)=h(x,y+1)$ for some $x,y$. If we modify the primitive recursion scheme as you suggest then $$ h(x,y+2)=g(x,h(x,y+1)) =g(x,h(x,y)) = h(x,y+1)$$ In other words if $h(x,y)=h(x,y+1)$ then $h(x,y)=h(x,z)$ for every $z>y$.

However there are primitive recursive functions that do not have this constraint, e.g. $$h(x,y):=\begin{cases}x+y & \text{if } y \text{ is even} \\ x+y-1 & \text{otherwise} \end{cases}$$