Define function $f(x,y,t)$ as the analytic continuation of the series
$$f(x,y,t)=\sum_{n,m\ge0}x^ny^mt^{nm}$$
This series definitely converges when all the arguments are small enough. I would like to understand the global properties of this function: presence of zeros, of poles or other singularities etc. In particular, I would like to **locate the poles** of this function w.r.t. to $y$ and **find the corresponding residues**.

Let us do a partial resummation $$f(x,y,t)=\sum_{n\ge0}\frac{x^n}{1-yt^n}$$ which is possible when say $|y|,|t|<1$.

Naively, from the above expression I expect that $f(x,y,t)$ has poles in $y$ for $y=t^{-n}, n\ge0$ with residues $$\operatorname{Res}_{y=t^{-n}}f(x,y,t)=-x^n/t^n$$

However, there is a problem here for me. I will first formulate is loosely and then give more accurate description.

## The problem

I. *Loose formulation.* Locations of poles of $f(x,y,t)$ are independent of $x$. Therefore, we don't expect them to change if $x$ changes. However, one can prove formula $f(x^{-1},y,t)=f(x,y^{-1},t)$ (modulo not-so-interesting terms). In the lhs we've only changed $x$, and hence we don't expect locations of poles to change. On the other hand in the rhs we've changed $y\to y^{-1}$ which causes poles to invert!

II. *More accurate description.*
By formally operating with the series one can deduce three following properties

- $f(x,y,t)=f(y,x,t)$, obvious symmetry.
- $f(x,y^{-1},t)=-yf(xt^{-1},y,t^{-1})$ rule to invert an argument.
$f(xt,yt,t)=(xyt)^{-1}(f(x,y,t)-(1-x)^{-1}-(1-y)^{-1}+1)$ scaling rule.

Using their combination one shows that

$$f(x^{-1},y,t)=f(x,y^{-1},t)+(1-y)^{-1}-(1-x)^{-1}$$

Again, as stated in the loose formulation, the poles in $y$ do not seem to agree between the lhs and rhs: in the lhs they are at the points $y=t^{-n}, n\ge0$ while at the rhs at the points $y=t^n, n\ge0$.

## Summary

I understand that most likely I messed up with divergent series somewhere, but I can't exactly find where. I would be grateful if somebody explained me where, but my main question is the following: **what are locations and residues of poles for function $f(x,y,t)$ wrt $y$-variable? Do they depend on $x$? If yes, how?**

## Appendix

Here I present formal derivation of properties 2, 3.

$$f(x, y^{-1},t)=\sum_{n\ge0}\frac{x^n}{1-y^{-1}t^n}=\sum_{n\ge0}\frac{x^nt^{-n}y}{yt^{-n}-1}=-yf(xt^{-1}, y, t^{-1})$$

And

$$f(xt, yt,t)=\sum_{n,m\ge0}{x^ny^mt^{nm+n+m}}=(xyt)^{-1}\sum_{n,m\ge0}x^{n+1}y^{m+1}t^{(n+1)(m+1)}=(xyt)^{-1}\left(\sum_{n,m\ge0}-\sum_{n\ge0,m=0}-\sum_{n=0,m\ge0}+\sum_{n=0,m=0}\right)x^{n}y^{m}t^{nm}=(xyt)^{-1}(f(x,y,t)-(1-x)^{-1}-(1-y)^{-1}+1)$$

## On Dmitry Vaintrob's answer

As far as I can the the factorization procedure suggested in this answer is not actually valid since the two factors has non-overlapping region of convergence. So, I'm still looking for a solution.