Suppose we have asymptotic series $$ f(x)\sim\sum_{n=0}^\infty a_n\, x^{-n},\quad x\to\infty $$ and $$ g(y)\sim\sum_{n=0}^\infty b_n\, y^{-n},\quad y\to\infty. $$ Is the product $$ \sum_{k=0}^\infty\sum_{\ell=0}^k a_\ell b_{k-\ell}\,x^{-\ell} \,y^{\ell-k},\quad x,y\to\infty $$ an asymptotic expansion for $f(x)g(y)$? If so, what is the order of the $k$th term?
I know this is true if $x=y$ in the sense that $$ f(x)g(x)\sim\sum_{k=0}^N\sum_{\ell=0}^k a_\ell b_{k-\ell} \,x^{-k}+\mathcal O\{x^{-N-1}\},\quad x\to\infty $$ is an asymptotic series.
I thought, perhaps, I could recast the problem in terms of polar coordinates $(x,y)\mapsto(\rho\cos\varphi,\rho\sin\varphi)$ such that $$ f(\rho\cos\varphi)g(\rho\sin\varphi)\sim\sum_{k=0}^N\sum_{\ell=0}^k a_\ell(\varphi) b_{k-\ell}(\varphi) \,\rho^{-k}+\mathcal O\{\rho^{-N-1}\},\quad \rho\to\infty. $$