Combinatorial Sums and Identities Involving Generalized Sum-of-Divisors Functions with Bounded Divisors

The class of Lambert series generating functions (LGFs) denoted by $L_{\alpha }(q)$ formally enumerate the generalized sum-of-divisors functions, ${\sigma }_{\alpha }(n)={\sum }_{d\mathrm{\mid }n}{d}^{\alpha }$, for all integers $n\ge 1$ and fixed real-valued parameters $\alpha \ge 0$. We prove new formulas expanding the higher-order derivatives of these LGFs. The results we obtain are combined to express new identities expanding the generalized sum-of-divisors functions. These new identities are expanded in the form of sums of polynomially scaled multiples of a related class of divisor sums depending on $n$ and $\alpha$.