Suppose $f: N \rightarrow R$, $g$ is a function of two variables $n \in N$ and $x \in R$, and $a$ is a real number, $a \in R$.
Definition: $f$  is a (first order) recursive function if $f(0) = a$ and for all $n$, $f(n+1) = g(n, f(n))$.

(First Order) Recursive Function Theorem: Given $a \in R$ and $g$ a function of two variables $n \in N$ and $x \in R$, there is a unique recursive function $f_{a,g}: N \rightarrow R$ with $f_{a,g}(0) = a$ and for all $n$, $f_{a,g}(n+1) = g(n, f_{a,g}(n))$.

Suppose $f: N \rightarrow R$, $g$ is a function of three variables $n \in N$, $x$ and $y \in R$, and $a$ is a real number, $a \in R$.
Definition: $f$  is a (second order) recursive function if $f(0) = a$ and for all $n$, $f(n+2) = g(n, f(n), f(n+1))$.

(Second Order) Recursive Function Theorem: Given $a \in R$ and $g$ is a function of three variables $n \in N$, $x$ and $y \in R$, there is a unique recursive function $f_{a,g}: N \rightarrow R$ with $f_{a,g}(0) = a$ and for all $n$, $f_{a,g}(n+1) = g(n, f_{a,g}(n), f_{a,g}(n+1))$.