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 (, z) ⁴, () ^12, () ⁴ and () ⁸ (2) ⁸ in terms of the Hurwitz class numbers, respectively. The proof is based on the theory of Jacobi forms.