New Recurrence Relations and Matrix Equations for Arithmetic Functions Generated by Lambert Series

We consider relations between the pairs of sequences, $(f,g_f)$, generated by the Lambert series expansions, $L_f(q) = \sum_{n \geq 1} f(n) q^n / (1-q^n)$,in$q$. In particular, we prove new forms of recurrence relations and matrix equations defining these sequences for all $n\in {\mathbb{Z}}^{+}$. 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, $(q;q{)}_{\mathrm{\infty }}$, 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, $\varphi (n)$, the M"obius function, $\mu (n)$, the sum of divisors functions, ${\sigma }_1(n)$ and ${\sigma }_{\alpha }(n)$, for $\alpha \ge 0$, and to Liouville’s lambda function, $\lambda (n)$.