Comparing two power series and extracting their coefficients

Question

I have a standard mathematical problem that I was wondering how to solve efficiently by mathematica. Here is the problem.

I have two power series expansions of a function F[x,y] = sum1 = sum2

A1 = (a/4)^2 (x^2 + y^2 + z^2); 
A2 =  (a/4)^3 x y z;
sum1 = Sum[A1^i * A2^j * P[i, j], {i, 0, 6}, {j, 0, 6}] /. {z -> -x - y}
sum2 = Sum[(a/4)^(i + j) (x^i y^j)/(i! j!) Q[i, j], {i, 0, 6}, {j, 0, 6}]

Basically I want to express one type of coefficients P[i,j] in terms of Q[i,j].

Answer 1

In sum2, the powers of x and y are less than or equal to 6, whereas in sum1, the powers go up much higher. For that reason, we'll have to truncate sum1.

Perhaps this is what you're looking for:

Thread /@ Thread[CoefficientList[sum1, {x, y}][[;; 7, ;; 7]] == CoefficientList[sum2, {x, y}]] // Flatten;
Solve[%, Flatten@Array[Q, {7, 7}, 0]]

This generates the set of coefficients for each power of $x^ny^m$, sets them equal to each other for the two sums (up to degree 6 in each variable), and then solves for the Q's in terms of the P's.