Kronecker theta function and a decomposition theorem for theta functions I

2020/12/03 by Zhiguo Liu, Liu, Zhi-Guo · 1 citation
Mathematics · #11F27 #11F37 #33D15 #Advanced Mathematical Identities #Analytic Number Theory Research #Complex Variables (math.CV) #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2012.01670

openalex publication_date 2020/12/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Kronecker theta function is a quotient of the Jacobi theta functions, which is also a special case of Ramanujan's ₁ψ₁ summation. Using the Kronecker theta function as building blocks, we prove a decomposition theorem for theta functions. This decomposition theorem is the common source of a large number of theta function identities. Many striking theta function identities, both classical and new, are derived from this decomposition theorem with ease. A new addition formula for theta functions is established. Several known results in the theory of elliptic theta functions due to Ramanujan, Weierstrass, Kiepert, Winquist and Shen among others are revisited. A curious trigonometric identities is proved.

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