  We propose a notion of iterating functions $f:X^{k}\rightarrow X$ in a way
that represents recurrence relations of the form
$a_{n+k}=f(a_{n},a_{n+1},...,a_{n+k-1})$. We define a function as
$n$-involutory when its $n$th iterate is the identity map, and discuss
elementary group-theoretic properties of such functions along with their
relation to cycles of their corresponding recurrence relations. Further, it is
shown that a function $f:X^{k}\rightarrow X$ that is 2-involutory in each of
its $k$ arguments (holding others fixed) is $(k+1)$-involutory.
