Does there exist a generating function for the rational numbers?

Question

Since the rationals are countable, you can list them in a sequence $(a_n)_{n\geq 0}$ such that each rational appears at least once in the sequence. Is there such a listing $(a_n)_{n \geq 0}$ for which $$\sum_{k = 0}^{\infty} a_kx^k =a_0 + a_1x + a_2x^2 + ...$$ has a closed form?

Answer 1

The following are just some preliminary thougts on how we could start and approach the problem, which are anyway too long for a comment.

Along the track of the Farey sequence, we could build a 2 variables generating function for all the rationals in $[0,1]$ as $$ f(x,y) = \sum\limits_{1\, \le \,n} {\sum\limits_{1 \le m \le n} {\left[ {\gcd (n,m) = 1} \right]{m \over n}x^{\,n} y^{\,m} } } $$ where $[P]$ denotes the Iverson bracket.

We can also construct the o.g.f. above following instead the steps of the Stern-Brocot tree $$ \eqalign{ & f_0 (x,y) = 0x + xy \cr & f_1 (x,y) = 0x + {1 \over 2}x^{\,2} y + xy \cr & f_2 (x,y) = 0x + {1 \over 3}x^{\,3} y + {1 \over 2}x^{\,2} y + {2 \over 3}x^{\,3} y^{\,2} + xy \cr & \quad \vdots \cr} $$

We know the algorithm for constructing the single terms of the function, but (to my knowledge) not a closed expression for it,
even changing to exponential g.f. or others.

Answer 2

I suggest using continued fractions to define an infinite-variable generating function (for rationals in the open unit interval) $$ G(x_1, x_2, \dots):=\sum_{1\leq k} \sum_{1\leq n_1, n_2,\dots,n_k} [0,n_1,n_2,\dots ,n_k]x_1^{n_1} x_2^{n_2}\dots x_k^{n_k} $$ (WARNING: I will be sloppy about indices, double representation of rationals by continued fractions, etc)

The first term in the above sum is $\log(1-x_1)$, and the second term is $$ \sum_{1\leq n_1, n_2} \frac{n_2}{n_1n_2+1}x_1^{n_1}x_2^{n_2} $$ which appears interesting enough and probably deserves a name. An idea to find a 'closed expression' is to express the coefficients as $$ \cfrac{1}{n_1+\cfrac{1}{n_2}}=\frac{1}{n_1}\sum\frac{(-1)^k}{(n_1n_2)^k} $$ which yields a sum involving lots of logarithms.

Another idea is to filter $G$ by sums of level $\ell$: $$ G(x_1, x_2, \dots)= \sum_{\ell=1}^\infty P(n_1, n_2, \dots ,n_\ell), $$ where $$ P(n_1, n_2, \dots n_\ell)= \sum_{1\leq k, n_1+n_2+\dots+n_k=\ell} [0,n_1, n_2, \dots n_k]x_1^{n_1}x_2^{n_k}\dots x_k^{n_k}. $$

If $x=x_1=x_2=\dots$ we get $$ G(x,x,\dots)=\sum_{1\leq \ell} 2^\ell x^\ell=\frac {1}{1-2x}, $$ since the sum of the $\ell$th level of the Farey tree is $2^\ell$.

It seems hopeless to find a closed form for $G$. On the other hand, note $$ \sum_{1\leq k} \sum_{1\leq n_1, n_2,\dots,n_k} x_1^{n_1} x_2^{n_2}\dots x_k^{n_k} =\frac{1}{1-x_1} \left(1+\frac{1}{1-x_2}\left(1+\frac{1}{1-x_3}\dots\right)\right) $$ $$ =\sum_{n=1}^\infty\prod_{k=1}^n \frac{1}{1-x_k} $$

Answer 3

Just some comments

How about using Cantor's pairing function ?

https://en.wikipedia.org/wiki/Pairing_function

Although this does not reduce the fractions, so you get them multiple times.

$$\frac{5}{10} = \frac{1}{2}$$

for instance both occur.

Or maybe more interesting is this Calkin-Wilf tree :

https://en.wikipedia.org/wiki/Calkin%E2%80%93Wilf_tree

Where all rationals occur only once !

Ofcourse this does not answer your question, it only lists the rationals.

Do you consider a lambert series as a closed form ??

I do not think there are closed forms in terms of simple functions.

The reason intuitively (my intuition) is this :

the n'th derivative of a 'simple' taylor series is often full of recursive properties.

But I do not see that happening for a sequence of all the rationals or even the rationals in a unit interval.

In fact I think even a taylor series containing the natural numbers only is usually not given by a closed form.

$$ \sum n x^{cn+d} $$ for fixed integer $c,d$

and results from series multisection can give rational functions but otherwise I do not see closed forms.

I will ask a question about that. Thanks for the inspiration.