Let $\displaystyle E(x)$ be an arbitrary even function and n be any positive integer.
Let
$\displaystyle H_i=\sum_{a}[E(\eta-\epsilon+i)+E(\eta-\epsilon-i)-E(\eta+\epsilon+i)-E(\eta+\epsilon-i)]$
$\displaystyle -\sum_{b}(\delta-\gamma)[E(\gamma+i)+E(\gamma-i)-E(i)-E(-i)].$
where the summations goes over the partitions of n
$\displaystyle (a)$ $\displaystyle n=\eta \theta + \epsilon \zeta$ with $\displaystyle \theta$ and $\displaystyle \zeta$ odd
$\displaystyle (b)$ $\displaystyle n=\gamma \delta$ with $\displaystyle \delta$ odd.
Given
$\displaystyle H_0(n)+2\sum_{i=1}^{\lfloor \sqrt{n-1} \rfloor}(-1)^i H_i(n-i^2)=0.$
Show that since this identity holds for all n, this implies that $\displaystyle H_0(n)=0$