2010/05/27 by Zhu Cao, Cao, Zhu
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.1005.5116
arxiv created 2010/05/27 · openalex publication_date 2010/05/27 · arxiv updated 2010/05/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we prove that there is a natural correspondence between product identities for theta functions and integer matrix exact covering systems. We show that since ℤn can be taken as the disjoint union of a lattice generated by n linearly independent vectors in ℤn and a finite number of its translates, certain products of theta functions can be written as linear combinations of other products of theta functions. We firstly give a general theorem to write a product of n theta functions as a linear combination of other products of theta functions. Many known identities for products of theta functions are shown to be special cases of our main theorem. Several entries in Ramanujan's notebooks as well as new identities are proved as applications, including theorems for products of three and four theta functions that have not been obtained by other methods.