I have a sum that is of the form $$S_{p}(x,y)=\sum_{n=1}^{p-1}\sum_{m=1}^{p-1-n} A_{n,m}(x,y),$$ where $A_{n,m}(x,y)$ is a monomial of the form $c_{n,m}x^ny^m$.
I wish to take a $p\rightarrow\infty$ limit and write the expression in terms of a bunch of infinite sums. One might argue (numerically) that given a few conditions (which are anyway required for the limit to be convergent), the constraint can straightaway be lifted and the answers are the same. However, the particular $A_{n,m}(x,y)$ I have possesses a few nice properties, which get destroyed while doing this approximation.
Is there a way to consistently take the limit and write the expression as a bunch of infinite sums?