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I am studying a sequence $(\alpha_{p,q})$ indexed by a pair of coprime integers; this sequence arises naturally in the study of a particular set of spaces in geometric topology, but unfortunately the definition is a bit too involved to describe here.

I am interested in finding the generating function for this sequence in closed form. (It turns out that there is a strong analogy between the sequence and a Lucas sequence and so this is why I expect that there is some chance of there being a closed form.) This brings me to my question:

What are some references for the study of generating functions of the form

$$ \sum_{\substack{p,q\\coprime}} \alpha_{p,q} x^p y^q $$

and are there any well-known examples of such generating functions which are known to have a closed form?

(I am aware of the MSE questions 3972644 and 3138912 which are relevant and suggest that generating functions of this form are "hard", I am hoping that the community here might know of some successes in the area which I might be able to adapt to my problem; I have also become interested in knowing about the general case even independent of my problem.)

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  • $\begingroup$ Silly suggestion, but you could define a slightly different generating function (that contains the same information) by setting $a_{m,n} := a_{p,q}$ where $p=m/\gcd(m,n)$, $q= n/\gcd(m,n)$, and it might be better behaved. $\endgroup$
    – Sam Hopkins
    Oct 11, 2021 at 13:20
  • $\begingroup$ @SamHopkins Thanks for the suggestion, I have had a bit of a think about that but unfortinately I don't think it will help in this case because my sequence is defined by a recursion down the Stern-Brocot tree (so when $ m $ and $ n $ are not coprime the coefficient $ a_{m,n} $ is not reachable at all). $\endgroup$
    – Alex Elzenaar
    Oct 13, 2021 at 3:16
  • $\begingroup$ Understood. The problem though is that lacunary generating functions tend to be poorly behaved. $\endgroup$
    – Sam Hopkins
    Oct 13, 2021 at 3:18

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