An interesting identity involving the abundancy index of divisors of odd perfect numbers

There is nothing too special about the identities, as we shall soon see.

Hereinafter, we let $q^k n^2$ be an odd perfect number with special prime $q$, and we assume that the Descartes-Frenicle-Sorli Conjecture that $k=1$ holds true. Also, we denote the abundancy index of the positive integer $x$ by $I(x)=sigma(x)/x$, where $sigma(x)$ is the classical sum of divisors of $x$.

We compute:

$$frac{d}{dq}(I(n^2) + I(q))=frac{d}{dq}bigg(frac{3q^2 + 2q + 1}{q(q+1)}bigg)=frac{d}{dq}bigg(3 - frac{q - 1}{q(q + 1)}bigg)=frac{d}{dq}bigg(frac{-1}{q+1}+frac{1}{q(q+1)}bigg)$$ $$=frac{d}{dq}bigg(frac{-2}{q+1}+frac{1}{q}bigg)=frac{2}{(q+1)^2}-frac{1}{q^2}= frac{2q^2 - (q+1)^2}{(q(q+1))^2}$$

But $$I(n^2) - I(q) = frac{2q}{q+1} - frac{q+1}{q} = frac{2q^2 - (q+1)^2}{q(q+1)}.$$

Similarly, we obtain:

$$frac{d}{dq}(I(n^2) - I(q))=frac{d}{dq}bigg(frac{q^2 - 2q - 1}{q(q+1)}bigg)=frac{d}{dq}(I(n^2) + I(q) - 2I(q))$$ $$=frac{d}{dq}bigg(3 - frac{q - 1}{q(q + 1)} - frac{2(q+1)}{q}bigg)=frac{d}{dq}bigg(frac{-1}{q+1}+frac{1}{q(q+1)}+frac{-2}{q}-2bigg)$$ $$=frac{d}{dq}bigg(frac{-2}{q+1}+frac{1}{q}+frac{-2}{q}-2bigg)=frac{d}{dq}bigg(frac{-2}{q+1}-frac{1}{q}-2bigg)=frac{2}{(q+1)^2}+frac{1}{q^2}$$ $$=frac{2q^2 + (q+1)^2}{(q(q+1))^2}.$$

But $$I(n^2)+I(q)=frac{2q}{q+1}+frac{q+1}{q}=frac{2q^2 + (q+1)^2}{q(q+1)}.$$

In the above computations, we have used the identity $$frac{1}{q(q+1)}=frac{1}{q}-frac{1}{q+1}.$$