When I studyied the representation of integers as sum of squares, I found that the most powerful tool is the Jacobi Triple Product, in fact this amazing identity allows us to find more useful equalities such as:
$$ \prod_{n\geq 1}(1+zq^{2n-1})(1+z^{-1}q^{2n-1})(1-q^{2n})=\sum_{n=-\infty}^{+\infty}z^nq^{n^2}$$ If we take $z\to 1$ we find an expression of $ S=\sum_{-\infty}^{+\infty} q^{n^2} $ and if we simplify $S^2$ and $S^4$ we can find the number of representation of integers as sum of $2$ squares and $4$ squares respectively which are denoted commonly by $r_2(n)$ and $r_4(n)$. My question: is there any formula which can help us find the number of representation of integers as sums of cubes?, for example a simplified expression for $$\sum_{n=0}^{+\infty}x^{n}y^{n^2}q^{n^3}$$
I mean by a "simplified expression" an expression which can help us finding an expansion of $S'^2$ or $S'^3$ where $S'=\sum_{n=0}^{+\infty} q^{n^3}$.
Thanks for your answers.
