Calculating some observable, I obtained the following-like converges sum $$ S = \sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\sum_{p=0}^{\min(n,k)} \frac{x^n}{n!} \frac{y^k}{k!} F(p), $$ where $F$ - some function which explicitly depends only on $p$ index. It is obvious that $$ S \neq e^x e^y \sum_{p=0}^{\infty} F(p). $$
But, probably, there are any methods how to manipulate such types of sums? Is it possible to somehow make a summation over $n$ and $k$, redefining the upper limit for the last sum and the function $F$?
To me it looks like, this expression cannot be significantly simplified further, but maybe I don't know some techniques?