Here's the relation:
if $n\ge j:$ then $$ \sigma(n,j,d) = d \cdot\left( \log j-\sigma\left(\frac{n}{j}, 1+d, d \right)\right)+\sigma(n, j+d, d )$$
And here's the terminating condition
if $n < j$ then $$ \sigma(n,j,d) = 0$$
Now, here's what I'm interested in:
$$ \frac{\partial}{\partial d}\sigma(n,j,d)=\text{ ?}$$
for some fixed positive whole numbers $n$ and $j$, and $d$ a real number such that $0<d\le 1$.
Is there a way to express this in a closed form?
(I hope this was the right tag)