Pentagonal number recurrence relations for $p(n)$

TL;DR

The paper extends Euler’s pentagonal-number recurrence for the partition function $p(n)$ to an infinite family indexed by $ u\ge0$. It constructs weight $2ν$ holomorphic modular forms $P_{ν}(τ)=[1/η,η]_{ν}$ via Rankin–Cohen brackets and expresses $p(n)$ through recurrences involving divisor sums $σ_{2ν-1}(n)$, Hecke traces $ ext{Tr}_{2ν}(n)$ of twisted quadratic Dirichlet-series values, and polynomials $g_{ν}(n,k)$. The ν=0 case recovers Euler’s recurrence, while special cases (notably ν∈{2,3,4,5,7} and {6,8,9,10,11,13}) yield explicit forms in terms of Eisenstein or cusp forms and, for ν=6, connect to Ramanujan’s τ-function via $ ext{Tr}_{12}(n)$; the general ν case introduces sums of twisted Dirichlet-series values through Petersson inner products. The work combines Poincaré-series representations of $p(n)$, unfolding techniques, and Rankin–Cohen theory to derive a rich, structurally explicit family of partition recurrences with meaningful arithmetic consequences, including Ramanujan-type congruences and a framework for computing weighted sums of Dirichlet-series values from partition data.

Abstract

We revisit Euler's partition function recurrence, which asserts, for integers $n\geq 1,$ that $$ p(n)=p(n-1)+p(n-2)-p(n-5)-p(n-7)+\dots = \sum_{k\in \mathbb{Z}\setminus \{0\}} (-1)^{k+1} p(n-ω(k)), $$ where $ω(m):=(3m^2+m)/2$ is the $m$th pentagonal number. We prove that this classical result is the $ν=0$ case of an infinite family of ``pentagonal number'' recurrences. For each $ν\geq 0,$ we prove for positive $n$ that $$ p(n)=\frac{1}{g_ν(n,0)}\left(α_ν\cdot σ_{2ν-1}(n)+ \mathrm{Tr}_{2ν}(n) +\sum_{k\in \mathbb{Z}\setminus \{0\}} (-1)^{k+1} g_ν(n,k)\cdot p(n-ω(k))\right), $$ where $σ_{2ν-1}(n)$ is a divisor function, $\mathrm{Tr}_{2ν}(n)$ is the $n$th weight $2ν$ Hecke trace of values of special twisted quadratic Dirichlet series, and each $g_ν(n,k)$ is a polynomial in $n$ and $k.$ The $ν=6$ case can be viewed as a partition theoretic formula for Ramanujan's tau-function, as we have $$ \mathrm{Tr}_{12}(n)=-\frac{33108590592}{691}\cdot τ(n). $$

Theorems & Definitions (40)

• Theorem 1.1
• Remark
• Corollary 1.2
• Remark
• Example
• Corollary 1.3
• Example
• Theorem 1.4
• Example
• Example