The class of Lambert series generating functions (LGFs) denoted by Lα(q) formally enumerate the generalized sum-of-divisors functions, σα(n)=∑d∣ndα
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, for all integers
n≥1
and fixed real-valued parameters
α≥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