Sums of four polygonal numbers: Precise formulas

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 $Z$-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 $\vartheta {(\tau ,z)}^4$, $\eta {\left(\tau \right)}^{12}$, $\eta {\left(\tau \right)}^4$ and $\eta {\left(\tau \right)}^8\eta {\left(2\tau \right)}^8$ in terms of Hurwitz class numbers, respectively. The proof is based on the theory of Jacobi forms.