A generating function for sums of $m$ squares

Power series can be multiplied together using the Cauchy product formula, essentially nothing more than the long multiplication algorithm for multiplying two polynomials. Further, it turns out that this product has a combinatorial interpretation. For powers of our generating function $$\vartheta_3(0,q)$$, this interpretation of the Cauchy product is essentially that the coefficient of $q^N$ in $$\vartheta_3^m(0,q)$$ counts the number of ways that $N$ can be written as the sum of $m$ squares. But let's be very specific as two what is being counted.

Consider the following representations of the number 5 as a sum of two squares:

$1^2 + 2^2$	$1^2 + (-2)^2$	$(-1)^2 + 2^2$	$(-1)^2 + (-2)^2$
$2^2 + 1^2$	$2^2 + (-1)^2$	$(-2)^2 + 1^2$	$(-2)^2 + (-1)^2$

From our point of view, these representations are distinct - the order of the squares in the sum matters and squaring negative integers is treated as distinct from squaring positive integers.

In fact, the coefficient of $q^5$ in $$\vartheta_3^2(0,q)$$ turns out to be 8, so the eight tabulated sums are the eight distinct ways of writing 5 as a sum of two squares.