How to prove the equality of power series below?

Question

Assume $$ F\left( x \right) := \sum_{n=-\infty}^{+\infty}{x^{\left( n+\frac{1}{2} \right) ^2}}, G\left( x \right) := \sum_{n=-\infty}^{+\infty}{x^{n^2}}, H\left( x \right) := \sum_{n=-\infty}^{+\infty}{\left( -1 \right) ^nx^{n^2}}, $$ I have to prove $$ \left( G\left( x \right) \right) ^2+\left( H\left( x \right) \right) ^2=2\left( G\left( x^2 \right) \right) ^2 $$ and $$ \left( G\left( x \right) \right) ^4=\left( F\left( x \right) \right) ^4+\left( H\left( x \right) \right) ^4. $$ After using Mathematica I find that it may have relation with EllipticTheta function. But it seems to be so difficult to me. Can anyone help me?

Answer 1

The coefficient of $x^k$ in $G(x)^2$ is the number of ways to write $k = m^2 + n^2$ where $m$ and $n$ are integers. The coefficient of $x^k$ in $H(x)^2$ is the number of ways to write $k = m^2 + n^2$ where $m +n$ is even minus the number of ways where $m+n$ is odd, thus $(-1)^k$ times the coefficient in $G(x)^2$. So the coefficient of $x^k$ in $G(x)^2 + H(x)^2$ is twice the number of ways to write $k = m^2 + n^2$ when $k$ is even, $0$ when $k$ is odd.

The coefficient of $x^k$ in $G(x^2)^2$ is $0$ if $k$ is odd (since $G(x^2)^2$ is an even function), and the coefficient of $x^{k/2}$ in $G(x)^2$ if $k$ is even. So that gives you your first equation.