Sums of four polygonal numbers: precise formulas

2024/05/23 by Li, Jialin, Wang, Haowu · 1 citation
#11D85 #11E25 #11F30 #11F50 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2405.14710

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

Cited by

• On restricted sums of four squares and Zhi-Wei Sun's x+24y conjecture

Related

• Conjectures of Sun about sums of polygonal numbers
• Representations of integers as sums of four polygonal numbers and partial theta functions
• Sums of Hurwitz Class Numbers and newform of weight 2 and level 49
• Universal sums of m-gonal numbers
• Numbers represented by restricted sums of four squares