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

TL;DR

This paper develops new formulas for Lambert series generating functions that enumerate generalized sum-of-divisors functions, leading to novel identities involving divisor sums with polynomial scaling.

Contribution

It introduces new formulas for derivatives of Lambert series and derives identities for generalized sum-of-divisors functions involving polynomially scaled divisor sums.

Findings

Derived formulas for higher-order derivatives of Lambert series

Established new identities for generalized sum-of-divisors functions

Expressed identities as sums involving polynomially scaled divisor sums

Abstract

The class of Lambert series generating functions (LGFs) denoted by $L_{\alpha}(q)$ formally enumerate the generalized sum-of-divisors functions, $\sigma_{\alpha}(n) = \sum_{d|n} d^{\alpha}$, for all integers $n \geq 1$ and fixed real-valued parameters $\alpha \geq 0$. 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 $\alpha$.