Let denote the recursive function of variables with Gödel number , where (1) is normally omitted. Then if is a partial recursive function, there exists an integer such that
where is Church's lambda notation. This is the variant most commonly known as Kleene's recursion theorem.
Another variant generalizes the first variant by parameterization, and is the strongest form of the recursion theorem. This form states that for each , there exists a recursive function of variables such that is a injection and if is a total function, then for all , ..., , and ,
Yet another and weaker variant of the recursion theorem guarantees the existence of a recursive function that is a fixed point for a recursive functional.