Sums of four polygonal numbers: precise formulas

TL;DR

This paper derives unified formulas for representing positive integers as sums of four polygonal numbers and squares, using Jacobi form theory, and proves several conjectures by Zhi-Wei Sun.

Contribution

It provides new explicit formulas for sums of four polygonal numbers and squares, connecting them to Hurwitz class numbers, and verifies several conjectures.

Findings

Formulas for sums of four polygonal numbers.

Formulas for restricted sums of four squares.

Verification of Zhi-Wei Sun's conjectures.

Abstract

In this paper we give unified formulas for the numbers of representations of positive integers as sums of four generalized $m$-gonal numbers, and as restricted sums of four squares under a linear condition, respectively. These formulas are given as $\mathbb{Z}$-linear combinations of Hurwitz class numbers. As applications, we prove several Zhi-Wei Sun's conjectures. As by-products, we obtain formulas for expressing the Fourier coefficients of $\vartheta(\tau,z)^4$, $\eta(\tau)^{12}$, $\eta(\tau)^4$ and $\eta(\tau)^8\eta(2\tau)^8$ in terms of Hurwitz class numbers, respectively. The proof is based on the theory of Jacobi forms.