A formula for the $r$-coloured partition function in terms of the sum of divisors function and its inverse

TL;DR

This paper derives a new explicit formula expressing the $r$-coloured partition function in terms of the sum of divisors function and provides an inverse relation, linking these two important number theoretic functions.

Contribution

It introduces a novel explicit formula for the $r$-coloured partition function involving divisor sums and their inverse, expanding the analytical tools for partition functions.

Findings

Derived a formula for $p_{-r}(n)$ using divisor sums and recursive sums.

Established an inverse relation expressing $\sigma(n)$ in terms of $p_{-r}(n)$.

Provides a new analytical connection between partition functions and divisor sums.

Abstract

Let $p_{-r}(n)$ denote the $r$-coloured partition function, and $\sigma(n)=\sum_{d|n}d$ denote the sum of positive divisors of $n$. The aim of this note is to prove the following $$ p_{-r}(n)=\theta(n)+\,\sum_{k=1}^{n-1}\frac{r^{k+1}}{(k+1)!} \sum_{\alpha_1\,= k}^{n-1} \, \sum_{\alpha_2\,= k-1}^{\alpha_1-1} \cdots \sum_{\alpha_k\, = 1}^{\alpha_{k-1}-1}\theta(n-\alpha_1) \theta(\alpha_1 -\alpha_2) \cdots \theta(\alpha_{k-1}-\alpha_k) \theta(\alpha_k) $$ where $\theta(n)=n^{-1}\, \sigma(n)$, and its inverse $$\sigma(n) = n\,\sum_{r=1}^n \frac{(-1)^{r-1}}{r}\, \binom{n}{r}\, p_{-r}(n). $$