For reasons that don’t matter here,
I want to estimate the power series coefficients
$t_{ij}$ for the rational function
$$T(x,y)= {(1+x)(1+y)over 1 x y(2+x+y+x y)}=sum_{i,j} t_{ij}x^iy^j$$
Using a method that I cannot justify, I get
highly accurate estimates when $i=j$ and highly inaccurate estimates when
$ij$ strays at all far from zero.
My questions are:
Q1) Why does my apparently illegitimate method work so well when $i=j$?
Q2) Why does the answer to Q1) not apply when $ineq j$ ?
(Of course, once the answer to Q1) is known, the answer to Q2) might be
selfevident.)
I’ll first present the method, then explain why I think it shouldn’t work,
then present the evidence that it works anyway when $i=j$, and then present
the evidence that it rapidly goes haywire when $ineq j$.
The Apparently Illegitimate Method:
Note that $t_{ij}=t_{ji}$, so we can limit ourselves to estimating
$t_{j+k,j}$ for $kge 0$.
I) Define
$$T_k(y)=sum_nt_{k+j,j}y^j$$
For example, a residue calculation gives
$$ T_0(y)= {1ysqrt{14y+2y^2+y^4}over ysqrt{14y+2y^2+y^4}} $$
It turns out that all of the $T_k$ share a branch point at $zetaapprox .2956$ and are analytic in the disc $r<zeta$.
II) Write
$$L_k=lim_{kmapsto zeta} T_k(y)sqrt{yzeta}$$.
Discover that $L_0approx 1.44641$ and $L_k=L_0zeta^{k/2}$.
III) Approximate
$$T_k(y)approx L_k/sqrt{yzeta}$$
IV) Expand the right hand side in a power series around $y=0$ and equate
coefficients to get
$$t_{ij}approx pm{L_0oversqrt{zeta}}pmatrix{1/2cr jcr}zeta^{(i+j)/2}
approx 2.66036 pmatrix{1/2cr jcr}zeta^{(i+j)/2}qquad(E1)$$
Remarks:

Obviously one could try to improve this approximation
at Step III by using more terms in the power series for $T_k$ at $y=zeta$.
This doesn’t seem to help, except when $k=0$, in which case the original approximation is already quite good.

For $kge 2$, $T_k(y)$ has a zero of
order $k1$ at the origin. Thus one could modify this method by approximating
$T_k(y)/(y^{k1})$ instead of $T_k(y)$
This yields
$$t_{ij}approx pm{2.66036}pmatrix{1/2cr 1i+2j}zeta^{(i+j)/2}qquad(E2)$$
(E2) is (much) better than (E1) in the range $ige 2j+1$, where it gets
exactly the correct value, namely zero. Otherwise, it seems neither systematically better nor worse.
Why Nothing Like This Should Work: The expansion of $T_k(y)$ at
$zeta$ contains nonzero terms of the form
$A_{i,j}(zetay)^j$ for all positive integers $j$. (I’m writing $i=j+k$ to
match up with the earlier indexing.) The truncation at Step III throws all
these terms away. Therefore the expansion around the origin in Step IV
ignores (among other things) the contribution of $A_{ij}$ to the estimate
for $t_{ij}$. So unless we can control the sizes of the $A_{ij}$, we
have absolutely no control over the quality of the estimate.
And in fact, even when $k=0$, the $A_{j,j}$ are not small.
For example, $t_{8,8}=8323$ and my estimate for $t_{8,8}$ is a
respectable $8962.52$. But $A_{8,8}$, which should have contributed to that
estimate and got truncated away, is equal to $58035$. It seems remarkable
that I can throw away multiple terms of that size and have the effects nearly cancel.
I’d like a conceptual explanation for this.
But When $i=j$, It Works Anyway:
and these get even better if you truncate just slightly farther out.
Why any explanation can’t be too general: