# asymptotic growth of coefficients of two-variable power series

Suppose you have a function of two variables, say $f(x,y)$, that is nice enough to equal a power series $\sum_{m,n} a_{m,n}x^my^n$ in some region about the origin.

Is there a rough asymptotic formula for $a_{m,n}$ based solely on $m$, $n$ and the singularities of $f$?

This question is essentially a two-dimensional version of the root test: if $f(x)=\sum_n a_nx^n$, and $r$ is the modulus of the smallest singularity of $f$, then $a_n\sim r^{-n}$.

Two simple-but-contrasting examples are $f_1(x,y)=\frac{1}{(1-2x)(1-3y)}$ with $a_{m,n}=2^m3^n$ and $f_2(x,y)=\frac{1}{1-xy}$ with $a_{m,n}=0$ unless $m=n$ and then $a_{m,m}=1$.

As a particular application, I am interested in approximating the coefficients of the rational function $\frac{1}{(1-x)^2(1-y)^2-xy}$, which is the generating function of an array of numbers of combinatorial objects I am studying.

• How are you defining "singularity of $f$"?
– zhw.
Commented May 4, 2017 at 22:13
• In the rational case, let's just say an $x,y$ pair so the numerator is non-zero and the denominator is zero. Commented May 4, 2017 at 22:52
• Well OK, but $\sim$ usually means asymptotic equivalence.
– zhw.
Commented May 5, 2017 at 16:36
• In case you aren't already aware of this resource (Twenty combinatorial examples of asymptotics derived from multivariate generating functions): arxiv.org/abs/math/0512548 Commented May 8, 2017 at 21:05
• @HughDenoncourt: That article looks like exactly what I was hoping for. I will study that for sure. Also, I see they have extended the article to an entire book. Many, many thanks for the reference! Commented May 9, 2017 at 1:11