  We prove new variants of the Lambert series factorization theorems studied by
Merca and Schmidt (2017) which correspond to a more general class of Lambert
series expansions of the form $L_a(\alpha, \beta, q) := \sum_{n \geq 1} a_n
q^{\alpha n-\beta} / (1-q^{\alpha n-\beta})$ for integers $\alpha, \beta$
defined such that $\alpha \geq 1$ and $0 \leq \beta < \alpha$. Applications of
the new results in the article are given to restricted divisor sums over
several classical special arithmetic functions which define the cases of
well-known, so-termed "ordinary" Lambert series expansions cited in the
introduction. We prove several new forms of factorization theorems for Lambert
series over a convolution of two arithmetic functions which similarly lead to
new applications relating convolutions of special multiplicative functions to
partition functions and $n$-fold convolutions of one of the special functions.
