A while back, someone posted (as an answer or a comment) a link to his paper showing that any Turing computable function had a closed form.
In both the comment and the paper itself, he freely and humorously admitted that he was stretching the definition of closed-form to the breaking point.
Does anyone know where I can find that paper and/or a general proof/method to convert Turing machines to closed form formulas?
I am not sure if I have ever seen Kleene post on this site, but his normal form theorem is the usual normal form for computable functions.
If we look at r.e. sets instead of at computable functions, the MRDP theorem has a corollary that for every r.e. set of naturals $A$ there is a multivariate polynomial $p(\bar x)$ with integer coefficients and integer variables, so that $A$ is the intersection of $\mathbb{N}$ with the range of $p$. In this sense $p$ is a closed form for the Turing machine that enumerates $A$.