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Abstract. We consider relations between the pairs of sequences, (f, g f ), generated by the Lambert series expansions, L f (q) = n≥1 f (n)q n /(1 − q n ), in q. In particular, we prove new forms of recurrence relations and matrix equations defining these sequences for all n ∈ Z + . The key ingredient to the proof of these results is given by the statement of Euler's pentagonal number theorem expanding the series for the infinite q-Pochhammer product, (q; q) ∞ , and for the first n terms of the partial products, (q; q) n , forming the denominators of the rational n th partial sums of L f (q). Examples of the new results given in the article include new exact formulas for and applications to the Euler phi function, φ(n), the Möbius function, µ(n), the sum of divisors functions, σ 1 (n) and σ α (n), for α ≥ 0, and to Liouville's lambda function, λ(n).

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We prove new variants of the Lambert series factorization theorems studied by which correspond to a more general class of Lambert series expansions of the form La(α, β; q) := n≥1 anq αn−β /(1− q αn−β ) for integers α, β defined such that α ≥ 1 and 0 ≤ β < α. 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.