Jtdylktuy

Is it possible to define the Cantor power series tuple-function of infinite degree $f_omega:Bbb N^{<omega}to Bbb N$?

I suspect at the moment this is an open problem but I don't know that anybody has given it any thought. If it can't be defined yet, should we expect the solution to be unique - i.e. does the Fueter–Pólya conjecture imply a) existence and b) uniqueness of such a power series, and c) is there reason to expect the definition to be degenerate?

It is a theorem of Rudolf Fueter and George Pólya that the Cantor pairing function given by $pi(x_1,x_2)=frac12left((x_1+x_2)^2+3x_1+x_2)right)$ is the sole quadratic pairing function from $pi:Bbb NtimesBbb NtoBbb N$, modulo exchanging $x_1$ for $x_2$.

The Fueter–Pólya conjecture is an open problem, claiming that the polynomial analogues of this function generated by induction of the same principle to $Bbb N^ktoBbb N$ are the only such polynomial tuple functions.

These are given by $$f_k(y)=sum_{n=1}^k binom{k-1+sum_{j=1}^{k}x_j}{k}$$

But I ask: Suppose we can take $ktoomega$. Then I assume that being a polynomial of infinite degree, this would be a power series. Is there reason to expect this power series tuple-function of infinite degree to be the only one, and is there hope to explicitly define it?

It seems relevant to mention that I am thinking that for every natural number this power series has finitely many nonzero terms.

My original motivation for this question was the following fact:

Suppose there exists a well-founded analogue of the Cantor tuple function $g:Bbb N^{alpha+1}toBbb N^{alpha}:alpha<omega$ which upon iteration reduces the last element of any given sequence $SinBbb N^{<omega}$ of integers by one until it is zero, then truncates the zero from $S$ before sequentially reducing the next term.

Then there is a representation of the Collatz graph which takes on precisely this form (given appropriate restrictions*). And it turns out the Collatz graph of any given number $x$ is in one to one correspondence with the unique power series representation of $x$ of the form $3^pcdot 2^{s_1}+3^{p-1}cdot 2^{s_2}+3^{p-2}cdot 2^{s_3}+ldots$, which makes a suitable definition of any given integer's Collatz sequence a candidate for the tuple-function of infinite degree.

*I'm waving away a fair bit of complexity with this statement.