Sums of four polygonal numbers: precise formulas

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 $\vartheta (\ddot{I}„,z{)}^4$, $\hat{I}\cdot (\ddot{I}„{)}^{12}$, $\hat{I}\cdot (\ddot{I}„{)}^4$ and $\hat{I}\cdot (\ddot{I}„{)}^8\hat{I}\cdot (2\ddot{I}„{)}^8$ in terms of Hurwitz class numbers, respectively. The proof is based on the theory of Jacobi forms.

Final version