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

## TL;DR

This paper introduces new recurrence relations and matrix equations for sequences generated by Lambert series, utilizing Euler's pentagonal number theorem, with applications to important arithmetic functions like Euler's phi, Möbius, and divisor functions.

## Contribution

It provides novel recurrence relations and matrix equations for Lambert series-generated sequences, expanding the analytical tools for arithmetic functions.

## Key findings

- Derived new exact formulas for Euler's phi function.
- Established recurrence relations for Möbius and divisor functions.
- Applied results to Liouville's lambda function.

## Abstract

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, $(q; q)_{\infty}$, 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, $\phi(n)$, the M\"obius function, $\mu(n)$, the sum of divisors functions, $\sigma_1(n)$ and $\sigma_{\alpha}(n)$, for $\alpha \geq 0$, and to Liouville's lambda function, $\lambda(n)$.