For $n\in\mathbb{N}$ and $q\in(0,1)$, define $$f_{n}(q):=\sum_{i_{1},i_{2},\dots,i_{n}=1}^{\infty}\frac{q^{i_1+i_2+\dots+i_n}}{(1-q^{i_1+i_2})(1-q^{i_2+i_3})\dots(1-q^{i_{n-1}+i_n})(1-q^{i_n+i_1})}.$$

I would like to known something about the asymptotic behavior of $f_{n}(q)$, as $n\to\infty$.

Some remarks on possible ways to find an answer: I believe that having, for instance, a recurrence rule for $f_{n}(q)$ or a generating function would be helpful. So far, however, I wasn't able to find anything. Perhaps there is a relation between $f_{n}(q)$ and Jacobi theta functions (or some other special functions from the $q$-world) which could be helpful as well.

Maybe the expression for $f_{n}(q)$ is familiar to someone. I would appreciate any information concerning the sequence (even if it does not yield the asymptotic formula). Many thanks.

For $n\in\mathbb{N}$ and $q\in(0,1)$, define $$f_{n}(q):=\sum_{i_{1},i_{2},\dots,i_{n}=1}^{\infty}\frac{q^{i_1+i_2+\dots+i_n}}{(1-q^{i_1+i_2})(1-q^{i_2+i_3})\dots(1-q^{i_{n-1}+i_n})(1-q^{i_n+i_1})}.$$

I would like to know something about the asymptotic behavior of $f_{n}(q)$, as $n\to\infty$.

Some remarks on possible ways to find an answer: I believe that having, for instance, a recurrence rule for $f_{n}(q)$ or a generating function would be helpful. So far, however, I wasn't able to find anything. Perhaps there is a relation between $f_{n}(q)$ and Jacobi theta functions (or some other special functions from the $q$-world) which could be helpful as well.

Maybe the expression for $f_{n}(q)$ is familiar to someone. I would appreciate any information concerning the sequence (even if it does not yield the asymptotic formula). Many thanks.